Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

Haitao Wan; Yongxiu Shi; Wei Liu

doi:10.1515/anona-2022-0199

Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

Wan, Haitao; Shi, Yongxiu; Liu, Wei 2022-01-01 00:00:00 1Introduction and main resultsThis presentation is to establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge–Ampère equation(1.1) det(D2u)=b(x)g(−u),u<0 in Ω and u=0 on ∂Ω,$$\begin{equation}\mbox{ det}(D^{2} u) =b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and }u=0 \mbox{ on }\partial\Omega,\end{equation}$$where Ω is a bounded, smooth and strictly convex domain in ℝN (N ≥ 2), andD2u(x)=(∂2u(x)∂xi∂xj)N×N$$D^{2}u(x)=\bigg(\frac{\partial^{2} u(x)}{\partial x_{i}\partialx_{j}}\bigg)_{N\times N}$$denotes the Hessian of u and D2u is the so called Monge–Ampère operator. The nonlinearity g satisfies(g1) g ∈ C1((0, ∞), (0, ∞)) is decreasing on (0, ∞) and lim t → 0 + g ( t ) = ∞ $ \lim\limits_{t\rightarrow0^{+}}g(t)=\infty $;(g2) there exist a constant γ>1 and some function f ∈ C1(0, a1)∩ C[0, a1) for a sufficiently small constant a1 > 0 such that−tg′(t)g(t):=γ+f(t) withlimt→0+f(t)=0,$$\frac{-tg'(t)}{g(t)}:=\gamma+f(t) \mbox{ with}\lim_{t\rightarrow0^{+}}f(t)=0,$$i.e.,g(t)=c0t−γexp∫ta1f(s)sds,c0=g(a1)a1γ,$$g(t)=c_{0}t^{-\gamma}\exp\left(\int_{t}^{a_{1}}\frac{f(s)}{s}ds\right),\,c_{0}=g(a_{1})a_{1}^{\gamma},$$where the function f satisfies(S1) f ≡ 0 on (0, a1] (or (S2) f(t)≠0, ∀ t ∈ (0, a] for some a ≤ a1).If (S2) holds in (g2) , then we suppose(g3) there exists θ ≥ 0 such thatlim t → 0 + t f ′ ( t ) f ( t ) = θ ≥ 0. $$\lim\limits_{t\rightarrow0^{+}}\frac{tf'(t)}{f(t)}=\theta\geq0.$$If θ=0 in (g3), then we further suppose(g4) there exist β ∈ ℝ+ and σ∈R$ \sigma\in \mathbb{R} $such thatlimt→0+(−lnt)βf(t)=σ.$$\lim_{t\rightarrow0^{+}}(-\ln t)^{\beta}f(t)=\sigma.$$The weight b satisfies(b1) b ∈ C∞(Ω) is positive in Ωand one of the following two conditions(b2) there exist k ∈ Λ, B0∈R$ B_{0}\in\mathbb{R} $and μ ∈ ℝ+ such thatb(x)=kN+1(d(x))(1+B0(d(x))μ+o((d(x))μ)),d(x)→0,$$b(x)=k^{N+1}(d(x))\big(1+B_{0}(d(x))^{\mu}+o((d(x))^{\mu})\big),\,d(x)\rightarrow0,$$where d(x)≔ dist(x, ∂Ω), x ∈ Ω, Λ denotes the set of all of positive monotonic functions in C1(0, δ0) ∩ L (0, δ0) which satisfylimt→0+ddt(K(t)k(t))=Dk≥0,K(t)=∫0tk(s)ds$$\lim_{t\rightarrow0^{+}}\frac{d}{dt}\bigg(\frac{K(t)}{k(t)}\bigg)=D_{k}\geq0,\,\,\,K(t)=\int_{0}^{t}k(s)ds$$and(b3) there exist L˜∈L$ \tilde{L}\in \mathcal{L} $, B0∈R$ B_{0}\in \mathbb{R} $and μ ∈ ℝ+ such thatb(x)=(d(x))−(N+1)L˜N(d(x))(1+B0(d(x))μ+o((d(x))μ)),d(x)→0,$$b(x)=(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\big(1+B_{0}(d(x))^{\mu}+o((d(x))^{\mu})\big),\,d(x)\rightarrow0,$$where L$ \mathcal {L} $denotes the set of all of positive functionsdefined on (0, t0] byL ~ ( t ) := c exp ⁡ ( ∫ t t 0 y ( s ) s d s ) , t ∈ ( 0 , t 0 ] , $$\tilde{L}(t):=c\exp\bigg(\int\limits_{t}^{t_{0}}\frac{y(s)}{s}ds\bigg),\,t\in(0,t_{0}],$$where c ∈ ℝ+, y ∈ C (0, t0] andlimt→0+y(t)=0$ \lim_{t\rightarrow0^{+}}y(t)=0 $.The set Λ in (b2) was first introduced by Cîrstea and Rădulescu [6]- [8] for non-decreasing functions and by Mohammed [33] for non-increasing functions to study the exact boundary behavior and uniqueness of boundary blow-up elliptic problems. When b satisfies (b2) with Dk > 0, we see by Lemma 3.1 (iii) -(iv) that b may be singular on the boundary with the index(1−Dk)(N+1)Dk>−(N+1).$$\frac{(1-D_{k})(N+1)}{D_{k}}>-(N+1).$$The condition (b3) implies that b is critical singular with the index −(N + 1).problem (1.1) has a wide range of applications in Riemannian geometry and optical physics and one important geometric application is to structure a Riemannian metric in Ω that is invariant under projective transformations. When g(t)=t−(N+2), t > 0 and b ≡ 1 in Ω, Nirenberg [38], Loewner and Nirenberg [31] for N = 2, Cheng and Yau [5] for N ≥ 2 studied the existence and uniqueness of solutions to problem (1.1). In particular, Cheng and Yau [5] showed that if Ω is convex and bounded but not necessarily strictly convex then problem (1.1) possesses a unique solution u∈C∞(Ω)∩C(Ωˉ)$ u\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $which is negative in Ω. When g(t)=t−γ (t > 0) with γ>1 and b∈C∞(Ωˉ)$ b\in C^{\infty}(\bar{\Omega}) $with b(x) > 0 for all x ∈ Ω, Lazer and McKenna [27] proved the existence and uniqueness of solutions to problem (1.1). Moreover, they also obtained the following global estimatec1(d(x))N+1N+γ≤u(x)≤c2(d(x))N+1N+γ,x∈Ω.$$c_{1}(d(x))^{\frac{N+1}{N+\gamma}}\leq u(x)\leqc_{2}(d(x))^{\frac{N+1}{N+\gamma}},\,x\in\Omega.$$When b satisfies (b1) and g:(0, ∞) → (0, ∞) is a non-increasing, smooth function, Mohammed [34] showed that problem (1.1) has a strictly convex solution u∈C∞(Ω)∩C(Ωˉ)$ u\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $if and only if the problem(1.2) det(D2u)=b(x) in Ω and u=0 on ∂Ω$$\begin{equation}\mbox{ det}(D^{2}u)=b(x) \mbox{ in }\Omega \mbox{ and }u=0 \mbox{ on }\partial\Omega\end{equation}$$has a strictly convex solution v∈C∞(Ω)∩C(Ωˉ)$ v\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $, where b may be singular or may vanish on ∂Ω. In particular, the author showed that(i1) if b∈C∞(Ωˉ)$ b\in C^{\infty}(\bar{\Omega}) $is positive in Ωˉ$ \bar{\Omega} $, then problem (1.1) has a strictly convex solution u∈C∞(Ω)∩C(Ωˉ)$ u\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $;(i2) if limt→0+g(t)=∞$ \lim_{t\rightarrow0^{+}}g(t)=\infty $,then problem (1.1) has a unique strictly convex solution u∈C∞(Ω)∩C(Ωˉ)$ u\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $and the solution usatisfiesc1ϕ(d(x))≤−u(x)≤c2ϕ(d(x)) and |∇u(x)|≤c2ϕ(d(x))d(x) near ∂Ω,$$c_{1}\phi(d(x))\leq-u(x)\leq c_{2}\phi(d(x)) \mbox{ and }|\nablau(x)|\leq c_{2}\frac{\phi(d(x))}{d(x)} \mbox{ near }\partial\Omega,$$where c1, c2 are positive constants and ϕ is uniquely determined by∫ 0 ϕ ( t ) ( G ( s ) ) − ( N + 1 ) d s = t , G ( t ) = ∫ t t ^ g ( s ) d s , t ∈ ( 0 , t ^ ) , t ^ ∈ ( 0 , ∞ ] ; $$\int\limits_{0}^{\phi(t)}(G(s))^{-(N+1)}ds=t,\,G(t)=\int\limits_{t}^{\hat{t}}g(s)ds,\,t\in(0,\hat{t}),\,\hat{t}\in (0, \infty];$$(i3) if b(x) ⩽ C(d (x))δ−N−1 for somepositive constants δ and C, then problem (1.2) has a strictly convex solution;(i4) if b(x)=C(d (x))−(N+1) for somepositive constant C, then problem (1.2) has no strictlyconvex solution.Later, Yang and Chang [47] extended the above results (i3) -(i4) to the following cases:(i5) if b(x) ⩽ C(d (x))−(N+1)(−ln d(x))−qnear ∂Ω for some q > N and C > 0, then problem (1.2) has a strictly convex solution;(i6) if b(x)=C(d (x))−(N+1)(−lnd(x))−N near ∂Ω for some C > 0, then problem (1.2) has no strictly convex solution.Let P∈C1(0,∞)$ \mathscr{P}\in C^{1}(0, \infty) $satisfy P′(t)<0$ \mathscr{P}'(t)<0 $and limt→0+P(t)=∞$ \lim_{t\rightarrow0^{+}}\mathscr{P}(t)=\infty $and define P(t)=∫t1P(s)ds$ \mathfrak{P}(t)=\int_{t}^{1}\mathscr{P}(s)ds $. Recently, under the hypothesis of (b1), Zhang and Du [49] obtain the following results:(i7) if b(x)≤P(d(x))$ b(x)\leq \mathscr{P}(d(x)) $near∂Ω and∫01(P(s))1/Nds<∞$ \int_{0}^{1}(\mathfrak{P}(s))^{1/N}ds<\infty $, then problem (1.2) has a strictly convex solution;(i8) if b(x)≥P(d(x))$ b(x)\geq \mathscr{P}(d(x)) $and∫01(P(s))1/Nds=∞$ \int_{0}^{1}(\mathfrak{P}(s))^{1/N}ds=\infty $, then problem (1.2) has no strictly convex solution.The above facts imply that problem (1.2) has a strictly convex solution if b satisfies (b1) andb(x)≤CkN+1(d(x)) near ∂Ω orb(x)≤C(d(x))−(N+1)L˜N(d(x)) near ∂Ω,$$b(x)\leq Ck^{N+1}(d(x)) \mbox{ near }\partial\Omega \mbox{ or}b(x)\leq C (d(x))^{-(N+1)}\tilde{L}^{N}(d(x)) \mbox{ near }\partial\Omega,$$where C is a positive constant, k ∈ Λ in (b2) and L˜∈L$ \tilde{L}\in\mathcal {L} $in (b3) with(1.3)∫ 0 t L ~ ( s ) s d s < ∞ . $$\begin{equation}\int\limits_{0}^{t}\frac{\tilde{L}(s)}{s}ds<\infty.\end{equation}$$In [28], Li and Ma studied the existence and the firstboundary behavior of the strictly convex solutions to problem (1.1) by using regularity theory and sub-supersolution method.In particular, when b ∈ C3(Ω) is positive in Ω and satisfies(b01) there exist k ∈ Λ and positiveconstants b1 and b2 such thatb1:=lim infd(x)→0b(x)kN+1(d(x))≤lim supd(x)→0b(x)kN+1(d(x))=:b2,$$b_{1}:=\liminf_{d(x)\rightarrow0}\frac{b(x)}{k^{N+1}(d(x))}\leq\limsup_{d(x)\rightarrow0}\frac{b(x)}{k^{N+1}(d(x))}=:b_{2},$$g satisfies (g1) and(g01)lim t → 0 + ( G ( t ) ) 1 / ( N + 1 ) ∫ 0 t ( G ( s ) ) − 1 / ( N + 1 ) d s = D g , G ( t ) = ∫ t a 1 g ( s ) d s for some a 1 > 0 , $$\lim\limits_{t\rightarrow0^{+}}(G(t))^{1/(N+1)}\int\limits_{0}^{t}(G(s))^{-1/(N+1)}ds=D_{g},\,\,\,G(t)=\int\limits_{t}^{a_{1}}g(s)ds\mbox{ for some}a_{1}>0,$$they showed that the unique strictly convex solution u to problem (1.1) satisfies1≤lim infd(x)→0−u(x)ψ(ϑ1K(d(x)))≤lim supd(x)→0−u(x)ψ(ϑ2K(d(x)))≤1,$$1\leq\liminf_{d(x)\rightarrow0}\frac{-u(x)}{\psi(\vartheta_{1}K(d(x)))}\leq\limsup_{d(x)\rightarrow0}\frac{-u(x)}{\psi(\vartheta_{2}K(d(x)))}\leq1,$$where ψ is uniquely determined by(1.4)∫ 0 ψ ( t ) ( ( N + 1 ) G ( s ) ) − 1 / ( N + 1 ) d s = t , $$\begin{equation}\int\limits_{0}^{\psi(t)}((N+1)G(s))^{-1/(N+1)}ds=t,\end{equation}$$ϑ1=(b1mˆ−(1−Dg−1(1−Dk)))1/(N+1) and ϑ2=(b2mˆ+(1−Dg−1(1−Dk)))1/(N+1)$$\vartheta_{1}=\bigg(\frac{b_{1}}{\hat{m}_{-}(1-D_{g}^{-1}(1-D_{k}))}\bigg)^{1/(N+1)}\mbox{ and }\vartheta_{2}=\bigg(\frac{b_{2}}{\hat{m}_{+}(1-D_{g}^{-1}(1-D_{k}))}\bigg)^{1/(N+1)}$$with(1.5)mˆ−:=maxxˉ∈∂ΩωN−1(xˉ) and mˆ+:=minxˉ∈∂ΩωN−1(xˉ),$$\begin{equation}\hat{m}_{-}:=\max_{\bar{x}\in\partial\Omega}\omega_{N-1}(\bar{x}) \mbox{ and }\hat{m}_{+}:=\min_{\bar{x}\in\partial\Omega}\omega_{N-1}(\bar{x}),\end{equation}$$whereωN−1(xˉ)=∏i=1N−1κi(xˉ)$$\omega_{N-1}(\bar{x})=\prod_{i=1}^{N-1}\kappa_{i}(\bar{x})$$denotes the (N − 1)-th curvature at xˉ$ \bar{x} $and κ1(xˉ),⋅⋅⋅,κN−1(xˉ)$ \kappa_{1}(\bar{x}),\cdot\cdot\cdot,\kappa_{N-1}(\bar{x}) $denote the principal curvatures of ∂Ω at xˉ$ \bar{x} $. In [51], Zhang showed that if b satisfies (b1) and (b01), g satisfies (g1) and(g02)lim t → 0 + ( ( g ( t ) ) 1 / N ) ′ ∫ 0 t ( g ( s ) ) − 1 / N d s = − C g , $$\lim\limits_{t\rightarrow0^{+}}((g(t))^{1/N})'\int\limits_{0}^{t}(g(s))^{-1/N}ds=-C_{g},$$and NDk + (1 + N)Cg > 1+N, then the unique strictly convex solution u to problem (1.1) satisfiesϑ31−Cg:=lim infd(x)→0−u(x)ϕg((K(d(x)))(N+1)/N)≤lim supd(x)→0−u(x)ϕg((K(d(x)))(N+1)/N)=:ϑ41−Cg,$$\vartheta_{3}^{1-C_{g}}:=\liminf_{d(x)\rightarrow0}\frac{-u(x)}{\phi_{g}((K(d(x)))^{(N+1)/N})}\leq\limsup_{d(x)\rightarrow0}\frac{-u(x)}{\phi_{g}((K(d(x)))^{(N+1)/N})}=:\vartheta_{4}^{1-C_{g}},$$where ϕg is uniquely determined by∫ 0 ϕ g ( t ) ( N g ( s ) ) − 1 / N d s = t , t > 0 , $$\int\limits_{0}^{\phi_{g}(t)}(Ng(s))^{-1/N}ds=t,\,t>0,$$ϑ3=((NN+1)Nb1mˆ−((1+N)Cg+NDk−1−N))1/N$$\vartheta_{3}=\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{b_{1}}{\hat{m}_{-}\big((1+N)C_{g}+ND_{k}-1-N\big)}\bigg)^{1/N}$$andϑ4=((NN+1)Nb2mˆ+((1+N)Cg+NDk−1−N))1/N.$$\vartheta_{4}=\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{b_{2}}{\hat{m}_{+}\big((1+N)C_{g}+ND_{k}-1-N\big)}\bigg)^{1/N}.$$Especially, if (b01) is replaced by the following condition(b02) there exist L˜∈L$ \tilde{L}\in\mathcal {L} $with (1.3) and positive constants b1 and b2 such thatb1:=lim infd(x)→0b(x)(d(x))−(N+1)L˜N(d(x))≤lim supd(x)→0b(x)(d(x))−(N+1)L˜N(d(x))=:b2,$$b_{1}:=\liminf_{d(x)\rightarrow0}\frac{b(x)}{(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))}\leq\limsup_{d(x)\rightarrow0}\frac{b(x)}{(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))}=:b_{2},$$Zhang [51] showed that the unique strictly convex solution u to problem (1.1) satisfiesϑ51−Cg≤lim infd(x)→0−u(x)ϕg(∫0d(x)L˜(s)sds)≤lim supd(x)→0−u(x)ϕg(∫0d(x)L˜(s)sds)≤ϑ61−Cg,$$\vartheta_{5}^{1-C_{g}}\leq\liminf_{d(x)\rightarrow0}\frac{-u(x)}{\phi_{g}\big(\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\big)}\leq\limsup_{d(x)\rightarrow0}\frac{-u(x)}{\phi_{g}\big(\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\big)}\leq\vartheta_{6}^{1-C_{g}},$$whereϑ5=(b1mˆ−N)1/N and ϑ6=(b2mˆ+N)1/N.$$\vartheta_{5}=\bigg(\frac{b_{1}}{\hat{m}_{-}N}\bigg)^{1/N} \mbox{ and }\vartheta_{6}=\bigg(\frac{b_{2}}{\hat{m}_{+}N}\bigg)^{1/N}.$$Then, Sun and Feng [43] and Li and Ma [29] generalized the above boundary behavior results to the case of the following Hessian equation for i=1, ···, NSi(D2u)=b(x)g(−u),u<0 in Ω and u=0 on ∂Ω,$$S_{i}(D^{2} u) =b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and }u=0\mbox{ on }\partial\Omega,$$whereS i ( D 2 u ) = S i ( λ 1 , ⋅ ⋅ ⋅ , λ N ) = ∑ 1 ≤ j 1 < ⋅ ⋅ ⋅ < j i ≤ N λ j 1 ⋅ ⋅ ⋅ λ j i $$S_{i}(D^{2}u)=S_{i}(\lambda_{1},\cdot\cdot\cdot,\lambda_{N})=\sum_{1\leqj_{1}<\cdot\cdot\cdot<j_{i}\leqN}\lambda_{j_{1}}\cdot\cdot\cdot\lambda_{j_{i}}$$and λ1, ···, λN are the eigenvalues of D2 u. Furthermore S0(λ) ≡ 1 for λ ∈ ℝN. Espically, Li and Ma [29] also studied the existence and uniqueness of viscosity solution to the problem. For related insights on the existence, regularity and asymptotic behavior of solutions to the Monge-Ampère equations, please refer to [4], [11], [19], [21]- [25], [30], [35]- [37], [44]- [45] and the references therein. When the Monge-Ampère operator (det(D2u)) is replaced by the Laplace operator (Δ), many papers have been dedicated to resolving existence, uniqueness and asymptotic behavior issues for solutions, please refer to [1]- [2], [10], [12]- [17], [26], [39], [46], [48], [50] and the references therein.In this paper, by making a complete and detailed analysis to some indexes in various cases, we establish the exact second boundary behavior of the unique strictly convex solution to problem (1.1), which is quite different from the first behavior of this solution. For all we know, in literature there aren't articles on the second boundary behavior of the strictly convex solution to problem (1.1).To our aims, we define the following subclasses of Λ and L$ \mathcal {L} $as follows:Λ 1 := { k ∈ Λ : lim t → 0 + t − 1 [ d d t ( K ( t ) k ( t ) ) − D k ] = E 1 , k } ; Λ 2 , β := { k ∈ Λ : lim t → 0 + ( − ln ⁡ t ) β [ d d t ( K ( t ) k ( t ) ) − D k ] = E 2 , k } ; L β := { L ~ ∈ L : lim t → 0 + ( − ln ⁡ t ) β y ( t ) = E 3 } , $$\begin{split}&\Lambda_{1}:=\bigg\{k\in\Lambda:\lim\limits_{t\rightarrow0^{+}}t^{-1}\bigg[\frac{d}{dt}\bigg(\frac{K(t)}{k(t)}\bigg)-D_{k}\bigg]=E_{1,k}\bigg\};\\&\Lambda_{2,\beta}:=\bigg\{k\in\Lambda:\lim\limits_{t\rightarrow0^{+}}(-\ln t)^{\beta}\bigg[\frac{d}{dt}\bigg(\frac{K(t)}{k(t)}\bigg)-D_{k}\bigg]=E_{2,k}\bigg\};\\&\mathfrak{L}_{\beta}:=\{\tilde{L}\in\mathcal {L}:\lim\limits_{t\rightarrow0^{+}}(-\ln t)^{\beta}y(t)=E_{3}\},\end{split}$$where β is a positive constant and the relation between L˜$ \tilde{L} $and y is given in (b3). Our results are summarized as follows and mˆ±$ \hat{m}_{\pm} $(given in Theorems 1.1-1.3) are defined by (1.5).Theorem 1.1Let b satisfy (b1) -(b2) with (γ+N)Dk>N + 1, g satisfy (g1) -(g2).(I) When (S1) holds (or (S2) and (g3) -(g4) hold with θ=0 in (g3)), we have(i) If k ∈ Λ1, and then the unique strictly convex solution u to problem (1.1) satisfies(1.6)ξ−ψ(K(d(x)))(1+C−(−lnd(x))−β+o((−lnd(x))−β))≤−u(x)≤ξ+ψ(K(d(x)))(1+C+(−lnd(x))−β+o((−lnd(x))−β)),d(x)→0,$$\begin{equation}\begin{split}&\xi_{-}\psi(K(d(x)))\big(1+C_{-}(-\ln d(x))^{-\beta}+o((-\lnd(x))^{-\beta})\big)\\&\leq -u(x) \leq \xi_{+}\psi(K(d(x)))\big(1+C_{+}(-\lnd(x))^{-\beta}+o((-\ln d(x))^{-\beta})\big),\,d(x)\rightarrow0,\end{split}\end{equation}$$where ψ is uniquely determined by (1.4) and(1.7)ξ±=(((γ+N)Dk−(N+1))mˆ±γ−1)−1/(γ+N),$$\begin{equation}\xi_{\pm}=\bigg(\frac{\big((\gamma+N)D_{k}-(N+1)\big)\hat{m}_{\pm}}{\gamma-1}\bigg)^{-1/(\gamma+N)},\end{equation}$$C ± = − ( ( γ + N ) D k N + 1 ) β C ± , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , 0 , i f ( S 1 ) h o l d s , $$C_{\pm}=\begin{cases}-\big(\frac{(\gamma+N)D_{k}}{N+1}\big)^{\beta}\mathfrak{C}_{\pm},\, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})}\,{ hold\, with}\,\theta=0,\\0,\,&\,{ if }\,\mathbf{(S_{1})} \,{ holds}\,,\end{cases}$$whereC ± = A ± γ + N , i f A + ≤ 0 a n d A − ≤ 0 , C + = A + γ + N m ^ − m ^ + a n d C − = A − γ + N m ^ + m ^ − , i f A + ≥ 0 a n d A − ≥ 0 , C + = A + γ + N m ^ − m ^ + a n d C − = A − γ + N , i f A + > 0 a n d A − < 0 , C + = A + γ + N a n d C − = A − γ + N m ^ + m ^ − , i f A + < 0 a n d A − > 0 $$\begin{align}\begin{cases}\mathfrak{C}_{\pm}=\frac{A_{\pm}}{\gamma+N},\, &\,{ if }\,A_{+}\leq0 \,{ and }\,A_{-}\leq0,\\\mathfrak{C}_{+}=\frac{A_{+}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\mathfrak{C}_{-}=\frac{A_{-}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,A_{+}\geq0 \,{ and }\,A_{-}\geq0,\\\mathfrak{C}_{+}=\frac{A_{+}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\mathfrak{C}_{-}=\frac{A_{-}}{\gamma+N},\,&\,{ if }\,A_{+}>0 \,{ and }\,A_{-}<0,\\\mathfrak{C}_{+}=\frac{A_{+}}{\gamma+N} \,{ and }\,\mathfrak{C}_{-}=\frac{A_{-}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,A_{+}<0 \,{ and }\,A_{-}>0\end{cases}\end{align}$$with(1.8)A±=(C0+1γ+Nlnmˆ±−1)σ,$$\begin{equation}A_{\pm}=\bigg(C_{0}+\frac{1}{\gamma+N}\ln\hat{m}_{\pm}^{-1}\bigg)\sigma,\end{equation}$$and(1.9)C0=(N+1)(1−Dk)((γ+N)Dk−(N+1))(γ−1)−(γ+N)−1ln(γ+N)Dk−(N+1)γ−1.$$\begin{equation}C_{0}=\frac{(N+1)(1-D_{k})}{\big((\gamma+N)D_{k}-(N+1)\big)(\gamma-1)}-(\gamma+N)^{-1}\ln\frac{(\gamma+N)D_{k}-(N+1)}{\gamma-1}.\end{equation}$$(ii) If k ∈ Λ2,β (β is the same as the one in(g4)), then (1.6) still holds, whereC ± = − C ^ ± , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , − D ^ ± , i f ( S 1 ) h o l d s , $$\begin{align}C_{\pm}=\begin{cases}-\hat{C}_{\pm},\, &\,{ if }\, \mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})}\,{ hold \,with}\,\theta=0,\\-\hat{D}_{\pm},\,&\,{ if }\,\mathbf{(S_{1})} \,{ holds},\end{cases}\end{align}$$andC ^ ± = A ± + B γ + N , i f A + + B ≤ 0 a n d A − + B ≤ 0 , C ^ + = A + + B γ + N m ^ − m ^ + a n d C ^ − = A − + B γ + N m ^ + m ^ − , i f A + + B ≥ 0 a n d A − + B ≥ 0 , C ^ + = A + + B γ + N m ^ − m ^ + a n d C ^ − = A − + B γ + N , i f A + + B > 0 a n d A − + B < 0 , C ^ + = A + + B γ + N a n d C ^ − = A − + B γ + N m ^ + m ^ − , i f A + + B < 0 a n d A − + B > 0 , $$\begin{align}\begin{cases}\hat{C}_{\pm}=\frac{\mathfrak{A}_{\pm}+\mathfrak{B}}{\gamma+N},\, &\,{ if }\,\mathfrak{A}_{+}+\mathfrak{B}\leq0 \,{ and }\,\mathfrak{A}_{-}+\mathfrak{B}\leq0,\\\hat{C}_{+}=\frac{\mathfrak{A}_{+}+\mathfrak{B}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\hat{C}_{-}=\frac{\mathfrak{A}_{-}+\mathfrak{B}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,\mathfrak{A}_{+}+\mathfrak{B}\geq0 \,{ and }\,\mathfrak{A}_{-}+\mathfrak{B}\geq0,\\\hat{C}_{+}=\frac{\mathfrak{A}_{+}+\mathfrak{B}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\hat{C}_{-}=\frac{\mathfrak{A}_{-}+\mathfrak{B}}{\gamma+N},\,&\,{ if }\,\mathfrak{A}_{+}+\mathfrak{B}>0 \,{ and }\,\mathfrak{A}_{-}+\mathfrak{B}<0,\\\hat{C}_{+}=\frac{\mathfrak{A}_{+}+\mathfrak{B}}{\gamma+N} \,{ and }\,\hat{C}_{-}=\frac{\mathfrak{A}_{-}+\mathfrak{B}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,\mathfrak{A}_{+}+\mathfrak{B}<0 \,{ and }\,\mathfrak{A}_{-}+\mathfrak{B}>0,\end{cases}\end{align}$$(1.10)D ^ + = B γ + N m ^ − m ^ + a n d D ^ − = B γ + N m ^ + m ^ − , i f B ≥ 0 , D ^ + = D ^ − = B γ + N , i f B < 0 $$\begin{equation}\begin{cases}\hat{D}_{+}=\frac{\mathfrak{B}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}} \,{ and }\,\hat{D}_{-}=\frac{\mathfrak{B}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}} ,\, &\,{ if }\,\mathfrak{B}\geq0,\\\hat{D}_{+}=\hat{D}_{-}=\frac{\mathfrak{B}}{\gamma+N},\,& \,{ if }\,\mathfrak{B}<0\end{cases}\end{equation}$$with(1.11)A ± = ( ( γ + N ) D k N + 1 ) β A ± a n d B = ( γ + N ) E 2 , k ( γ + N ) D k − ( N + 1 ) , $$\begin{equation}\mathfrak{A}_{\pm}=\bigg(\frac{(\gamma+N)D_{k}}{N+1}\bigg)^{\beta}A_{\pm}\,{ and }\,\mathfrak{B}=\frac{(\gamma+N)E_{2,k}}{(\gamma+N)D_{k}-(N+1)},\end{equation}$$where A ± are given by (1.8).(II) When (S2) and (g3) hold with θ > 0 in (g3) and k ∈ Λ2,β, then the unique strictly convexsolution u to problem (1.1) satisfies (1.6) with C±=−Dˆ±,$ C_{\pm}=-\hat{D}_{\pm}, $where Dˆ±$ \hat{D}_{\pm} $are given by(1.10).Corollary 1.1In Theorem 1.1, if Ω is a ball with radius R and center x0, then(I) When (S1) holds (or(S2) and (g3) -(g4) hold with θ = 0 in (g3)), we have(i) If k ∈ Λ1, then the unique strictly convex solution u to problem (1.1) satisfies(1.12)−u(x)=ξR1ψ(K(R−r))(1+CR1(−ln(R−r))β+o((−ln(R−r))β)),r→R,$$\begin{equation}-u(x)=\xi_{R}^{1}\psi(K(R-r))\big(1+C_{R}^{1}(-\ln(R-r))^{\beta}+o((-\ln (R-r))^{\beta})\big),\,r\rightarrow R,\end{equation}$$where r=∣x − x0∣,(1.13)ξR1=((γ+N)Dk−(N+1)(γ−1)RN−1)−1/(γ+N)$$\begin{equation}\xi_{R}^{1}=\bigg(\frac{(\gamma+N)D_{k}-(N+1)}{(\gamma-1)R^{N-1}}\bigg)^{-1/(\gamma+N)}\end{equation}$$andC R 1 = C ^ R 1 , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , 0 , i f ( S 1 ) h o l d s , $$C_{R}^{1}=\begin{cases}\hat{C}_{R}^{1},\, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})}\,{ hold\, with}\,\theta=0,\\0,\,&\,{ if }\, \mathbf{(S_{1})} \,{ holds},\,\end{cases}$$where(1.14)CˆR1=−((γ+N)DkN+1)β(C0+N−1γ+NlnR)σγ+N$$\begin{equation}\hat{C}_{R}^{1}=-\bigg(\frac{(\gamma+N)D_{k}}{N+1}\bigg)^{\beta}\frac{\big(C_{0}+\frac{N-1}{\gamma+N}\lnR\big)\sigma}{\gamma+N}\end{equation}$$and C0 is given by (1.9).(ii) If k ∈ Λ2,β (β is the same as the one in (g4)), then (1.12) still holds, whereC R 1 = C ^ R 1 − E 2 , k ( γ + N ) D k − ( N + 1 ) , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , − E 2 , k ( γ + N ) D k − ( N + 1 ) , i f ( S 1 ) h o l d s , $$C_{R}^{1}=\begin{cases}\hat{C}_{R}^{1}-\frac{E_{2,k}}{(\gamma+N)D_{k}-(N+1)},\, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})}\,{ hold\, with}\,\theta=0,\\-\frac{E_{2,k}}{(\gamma+N)D_{k}-(N+1)},\,&\,{ if }\,\mathbf{(S_{1})} \,{ holds}\,,\end{cases}$$where CˆR1$ \hat{C}_{R}^{1} $is given by (1.14) and C0 isgiven by (1.9).(II) When (S2) and (g3) hold with θ > 0 in (g3) and k ∈ Λ2,β, then the unique strictly convex solution u to problem (1.1) satisfies (1.12), whereCR1=−E2,k(γ+N)Dk−(N+1).$$C_{R}^{1}=-\frac{E_{2,k}}{(\gamma+N)D_{k}-(N+1)}.$$Theorem 1.2Let b satisfy (b1) -(b2) with μ ∈ (0, 1), g satisfy (g1) -(g2) with (1−μ)(γ+N)Dk>N + 1, and if (S2) holds in (g2),we further suppose that (g3) with θ > 0 and the following hold(1.15)θ ( N + 1 ) μ − N > γ > N + 1 ( 1 − μ ) D k − N , i f D k ∈ ( 0 , 1 ) , θ ( N + 1 ) − N > γ > max { N + 1 ( 1 − μ ) D k − N , 2 N + 1 N } a n d μ D k ∈ ( 0 , 1 ) , i f D k ∈ [ 1 , ∞ ) . $$\begin{equation}\begin{cases}\frac{\theta(N+1)}{\mu}-N>\gamma>\frac{N+1}{(1-\mu)D_{k}}-N,\, &\,{ if }\,D_{k}\in(0, 1),\\\theta(N+1)-N>\gamma>\max\big\{\frac{N+1}{(1-\mu)D_{k}}-N,\,\,\,\frac{2N+1}{N}\big\} \,{ and }\,\mu D_{k}\in (0, 1),\,&\,{ if }\,D_{k}\in[1, \infty).\end{cases}\end{equation}$$If k ∈ Λ1, then the unique strictly convex solution u to problem (1.1) satisfies(1.16)ξ−ψ(K(d(x)))(1+C˜−(d(x))μ+o((d(x))μ))≤−u(x)≤ξ+ψ(K(d(x)))(1+C˜+(d(x))μ+o((d(x))μ)),d(x)→0,$$\begin{equation}\begin{split}&\xi_{-}\psi(K(d(x)))\big(1+\tilde{C}_{-}(d(x))^{\mu}+o((d(x))^{\mu})\big)\\&\leq -u(x)\leq\xi_{+}\psi(K(d(x)))\big(1+\tilde{C}_{+}(d(x))^{\mu}+o((d(x))^{\mu})\big),\,d(x)\rightarrow0,\end{split}\end{equation}$$where ψ is uniquely determined by (1.4), ξ ± are given by (1.7) andC ~ ± = B 0 ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) H + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ≥ 0 , C ~ + = B 0 ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) H m ^ − m ^ + + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 < 0 , C ~ − = B 0 ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) H m ^ + m ^ − + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 < 0 $$\begin{cases}\tilde{C}_{\pm}=\frac{B_{0}(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}{\mathfrak{H}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}\geq0,\\\tilde{C}_{+}=\frac{B_{0}(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}{\mathfrak{H}\frac{\hat{m}_{-}}{\hat{m}_{+}}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}<0,\\\tilde{C}_{-}=\frac{B_{0}(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}{\mathfrak{H}\frac{\hat{m}_{+}}{\hat{m}_{-}}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}<0\end{cases}$$withH=((1−μ)(γ+N)Dk−(N+1))(N(N+1)+μ(N−1)(γ+N)Dk)+μ(γ+N)Dk((N−2)(N+1)+μ(N−1)(γ+N)Dk)+μ(1−μ)(γ+N)2Dk2.$$\begin{split}\mathfrak{H}&=\big((1-\mu)(\gamma+N)D_{k}-(N+1)\big)\big(N(N+1)+\mu(N-1)(\gamma+N)D_{k}\big)\\&+\mu(\gamma+N)D_{k}\big((N-2)(N+1)+\mu(N-1)(\gamma+N)D_{k}\big)+\mu(1-\mu)(\gamma+N)^{2}D_{k}^{2}.\end{split}$$Corollary 1.2In Theorem 1.2, if Ω is a ball with radius R and center x0, then the unique strictly convex solution u to problem (1.1) satisfies−u(x)=ξR1ψ(K(R−r))(1+CR2(R−r)μ+o((R−r)μ)),r→R,$$-u(x)=\xi_{R}^{1}\psi(K(R-r))\big(1+C_{R}^{2}(R-r)^{\mu}+o((R-r)^{\mu})\big),\,r\rightarrowR,$$whereξR1$ \xi_{R}^{1} $is given by (1.13), r=∣x − x0∣ andCR2=B0(N+1)((γ+N)Dk−(N+1))H+γ(N+1)((γ+N)Dk−(N+1)).$$C_{R}^{2}=\frac{B_{0}(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}{\mathfrak{H}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}.$$Remark 1.1In Theorem 1.2, if we replace k ∈ Λ1 byk∈Λ′:={k∈Λ:limt→0+t−μ(ddt(K(t)k(t))−Dk)=E1,k}$$k\in\Lambda':=\bigg\{k\in\Lambda:\lim_{t\rightarrow0^{+}}t^{-\mu}\bigg(\frac{d}{dt}\bigg(\frac{K(t)}{k(t)}\bigg)-D_{k}\bigg)=E_{1,k}\bigg\}$$and other conditions still hold, then (1.16) holds, whereC ~ ± = ( N + 1 ) [ B 0 ( ( γ + N ) D k − ( N + 1 ) ) − ( γ + N ) E 1 , k ] H + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ( ( γ + N ) D k − ( N + 1 ) ) ≥ ( γ + N ) E 1 , k , C ~ + = ( N + 1 ) [ B 0 ( ( γ + N ) D k − ( N + 1 ) ) − ( γ + N ) E 1 , k ] H m ^ − m ^ + + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ( ( γ + N ) D k − ( N + 1 ) ) < ( γ + N ) E 1 , k , C ~ − = ( N + 1 ) [ B 0 ( ( γ + N ) D k − ( N + 1 ) ) − ( γ + N ) E 1 , k ] H m ^ + m ^ − + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ( ( γ + N ) D k − ( N + 1 ) ) < ( γ + N ) E 1 , k . $$\begin{cases}\tilde{C}_{\pm}=\frac{(N+1)\big[B_{0}\big((\gamma+N)D_{k}-(N+1)\big)-(\gamma+N)E_{1,k}\big]}{\mathfrak{H}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},&\,{ if }\,B_{0}((\gamma+N)D_{k}-(N+1))\geq(\gamma+N)E_{1,k},\\\tilde{C}_{+}=\frac{(N+1)\big[B_{0}\big((\gamma+N)D_{k}-(N+1)\big)-(\gamma+N)E_{1,k}\big]}{\mathfrak{H}\frac{\hat{m}_{-}}{\hat{m}_{+}}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}((\gamma+N)D_{k}-(N+1))<(\gamma+N)E_{1,k},\\\tilde{C}_{-}=\frac{(N+1)\big[B_{0}\big((\gamma+N)D_{k}-(N+1)\big)-(\gamma+N)E_{1,k}\big]}{\mathfrak{H}\frac{\hat{m}_{+}}{\hat{m}_{-}}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}((\gamma+N)D_{k}-(N+1))<(\gamma+N)E_{1,k}.\end{cases}$$When b is critical singular on ∂Ω, we have the following second boundary behavior.Theorem 1.3Let b satisfy (b1) and (b3), g satisfy (g1) -(g2), and if (S2) holds in (g2), we further suppose that (g3) -(g4) hold with θ = 0 in (g3) and with β ∈ (0, 1] in (g4). If L˜∈Lβ$ \tilde{L}\in \mathfrak{L}_{\beta} $with(1.17)E3<0,β∈(0,1),−1,β=1,$$\begin{equation}E_{3}<\begin{cases} 0,\,&\beta\in(0, 1),\\-1,\, &\beta=1,\end{cases}\end{equation}$$then the unique strictly convex solution u to problem (1.1) satisfiesη − ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + C − ∗ M ( d ( x ) ) + o ( M ( d ( x ) ) ) ) ≤ − u ( x ) ≤ η + ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + C + ∗ M ( d ( x ) ) + o ( M ( d ( x ) ) ) ) , d ( x ) → 0 , $$\begin{split}&\eta_{-}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+C_{-}^{*}\mathfrak{M}(d(x))+o(\mathfrak{M}(d(x)))\big)\\&\leq -u(x)\leq\eta_{+}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+C_{+}^{*}\mathfrak{M}(d(x))+o(\mathfrak{M}(d(x)))\big),\,d(x)\rightarrow0,\end{split}$$where ψ is uniquely determined by (1.4),η±=((NN+1)Nγ+Nγ−1mˆ±)−1/(γ+N),$$\begin{align}\eta_{\pm}=\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\gamma+N}{\gamma-1}\hat{m}_{\pm}\bigg)^{-1/(\gamma+N)},\end{align}$$M ( d ( x ) ) = ( − ln ⁡ ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ) − β , $$\mathfrak{M}(d(x))=\bigg(-\ln\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\bigg)^{-\beta},$$andC ± ∗ = D ± , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , 0 , i f ( S 1 ) h o l d s , $$\begin{align}C_{\pm}^{*}=\begin{cases} \mathfrak{D}_{\pm},\,&\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})} \,{ hold\, with}\, \theta=0,\\0,\, &\,{ if }\,\mathbf{(S_{1})} \,{ holds},\end{cases}\end{align}$$whereD ± = X ± γ + N , i f X + ≥ 0 a n d X − ≥ 0 , D + = X + γ + N m ^ − m ^ + a n d D − = X − γ + N m ^ + m ^ − , i f X + ≤ 0 a n d X − ≤ 0 , D + = X + γ + N a n d D − = X − γ + N m ^ + m ^ − , i f X + > 0 a n d X − < 0 , D + = X + γ + N m ^ − m ^ + a n d D − = X − γ + N , i f X + < 0 a n d X − > 0 $$\begin{cases}\mathfrak{D}_{\pm}=\frac{\mathfrak{X_{\pm}}}{\gamma+N},\,& \,{ if }\,\mathfrak{X}_{+}\geq0 \,{ and }\,\mathfrak{X}_{-}\geq0,\\\mathfrak{D}_{+}=\frac{\mathfrak{X}_{+}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\mathfrak{D}_{-}=\frac{\mathfrak{X}_{-}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,\mathfrak{X}_{+}\leq0 \,{ and }\,\mathfrak{X}_{-}\leq0,\\\mathfrak{D}_{+}=\frac{\mathfrak{X}_{+}}{\gamma+N}\,{ and }\,\mathfrak{D}_{-}=\frac{\mathfrak{X}_{-}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,\mathfrak{X}_{+}>0 \,{ and }\,\mathfrak{X}_{-}<0,\\\mathfrak{D}_{+}=\frac{\mathfrak{X}_{+}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\mathfrak{D}_{-}=\frac{\mathfrak{X}_{-}}{\gamma+N},\,&\,{ if }\,\mathfrak{X}_{+}<0 \,{ and }\,\mathfrak{X}_{-}>0\end{cases}$$withX±=(γ+N)−1(N+1γ−1+ln((NN+1)Nγ+Nγ−1)+lnmˆ±)σ.$$\mathfrak{X}_{\pm}=(\gamma+N)^{-1}\bigg(\frac{N+1}{\gamma-1}+\ln\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\gamma+N}{\gamma-1}\bigg)+\ln\hat{m}_{\pm}\bigg)\sigma.$$Corollary 1.3In Theorem 1.3, if Ω is a ball with radius R and center x0, then the unique strictly convex solution u to problem (1.1) satisfies− u ( x ) = η R 1 ψ ( ( ∫ 0 R − r L ( s ) s d s ) N N + 1 ) ( 1 + C R 3 M ( R − r ) + o ( M ( R − r ) ) ) , r → R , $$-u(x)=\eta_{R}^{1}\psi\bigg(\bigg(\int\limits_{0}^{R-r}\frac{L(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+C_{R}^{3}\mathfrak{M}(R-r)+o(\mathfrak{M}(R-r))\big),\,r\rightarrowR,$$whereηR1=((NN+1)Nγ+N(γ−1)RN−1)−1/(γ+N),r=|x−x0|$$\eta_{R}^{1}=\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\gamma+N}{(\gamma-1)R^{N-1}}\bigg)^{-1/(\gamma+N)},\,r=|x-x_{0}|$$andCR3=(N+1γ−1+ln((NN+1)Nγ+Nγ−1)−(N−1)lnR)σ(γ+N)2.$$C_{R}^{3}=\frac{\big(\frac{N+1}{\gamma-1}+\ln\big(\big(\frac{N}{N+1}\big)^{N}\frac{\gamma+N}{\gamma-1}\big)-(N-1)\lnR\big)\sigma}{(\gamma+N)^{2}}.$$Remark 1.2In Theorem 1.3, we obtain by Lemma 3.2 (i) that if L˜∈Lβ$ \tilde{L}\in \mathfrak{L}_{\beta} $with (1.17), then (1.3) holds. But, when L˜∈Lβ$ \tilde{L}\in\mathfrak{L}_{\beta} $with β > 1, by a direct calculation we see that(1.18)∫ 0 t 0 L ~ ( s ) s d s = ∞ . $$\begin{equation}\int\limits_{0}^{t_{0}}\frac{\tilde{L}(s)}{s}ds=\infty.\end{equation}$$Since limt→0+(−lnt)βy(t)=E3$ \lim_{t\rightarrow0^{+}}(-\ln t)^{\beta}y(t)=E_{3} $, for any ε > 0 we can choose a small enough constantt˜<min{t0,1}$ \tilde{t}<\min\{t_{0}, 1\} $such thatE3−ε(−lnt)β≤y(t),t∈(0,t˜].$$\frac{E_{3}-\varepsilon}{(-\ln t)^{\beta}}\leq y(t),\,t\in(0,\tilde{t}].$$A simple calculation shows thatexp ⁡ ( ∫ t t ~ y ( s ) s d s ) ≥ exp ⁡ ( ∫ t t ~ E 3 − ε s ( − ln ⁡ s ) β d s ) = exp ⁡ ( E 3 − ε β − 1 ( ( − ln ⁡ t ~ ) 1 − β − ( − ln ⁡ t ) 1 − β ) ) . $$\begin{align}\exp\bigg(\int\limits_{t}^{\tilde{t}}\frac{y(s)}{s}ds\bigg)\geq\exp\bigg(\int\limits_{t}^{\tilde{t}}\frac{E_{3}-\varepsilon}{s(-\lns)^{\beta}}ds\bigg)=\exp\bigg(\frac{E_{3}-\varepsilon}{\beta-1}\big((-\ln\tilde{t})^{1-\beta}-(-\ln t)^{1-\beta}\big)\bigg).\end{align}$$Moreover, we have(1.19)lim t → 0 + exp ⁡ ( ∫ t t ~ E 3 − ε s ( − ln ⁡ s ) β d s ) = lim t → 0 + exp ⁡ ( E 3 − ε β − 1 ( ( − ln ⁡ t ~ ) 1 − β − ( − ln ⁡ t ) 1 − β ) ) = exp ⁡ ( E 3 − ε β − 1 ( − ln ⁡ t ~ ) 1 − β ) > 0. $$\begin{equation}\begin{split}\lim\limits_{t\rightarrow0^{+}}\exp\bigg(\int\limits_{t}^{\tilde{t}}\frac{E_{3}-\varepsilon}{s(-\lns)^{\beta}}ds\bigg)&=\lim\limits_{t\rightarrow0^{+}}\exp\bigg(\frac{E_{3}-\varepsilon}{\beta-1}\big((-\ln\tilde{t})^{1-\beta}-(-\lnt)^{1-\beta}\big)\bigg)\\&=\exp\bigg(\frac{E_{3}-\varepsilon}{\beta-1}(-\ln\tilde{t})^{1-\beta}\bigg)>0.\end{split}\end{equation}$$It follows by the definition of L˜$ \tilde{L} $in (b3) that there exists a positive constant C˜$ \tilde{C} $such that(1.20)C ~ exp ⁡ ( ∫ t t 0 E 3 − ε s ( − ln ⁡ s ) β d s ) ≤ L ~ ( t ) , t ∈ ( 0 , t 0 ] . $$\begin{equation}\tilde{C}\exp\bigg(\int\limits_{t}^{t_{0}}\frac{E_{3}-\varepsilon}{s(-\lns)^{\beta}}ds\bigg)\leq \tilde{L}(t),\,t\in(0, t_{0}].\end{equation}$$Combining (1.19) with (1.20), we obtain (1.18) holds.Remark 1.3If y ( t ) = ± ( − ln ⁡ t ) − 1 , 0 < t ≤ t ~ < min { t 0 , 1 } $ y(t)=\pm(-\ln t)^{-1},\,0<t\leq \tilde{t}< \min\{t_{0}, 1\} $in (b3), then we haveL ~ ( t ) = exp ⁡ ( ∫ t t 0 y ( s ) s d s ) = exp ⁡ ( ∫ t ~ t 0 y ( s ) s d s ) exp ⁡ ( ± ∫ t t ~ d s s ( − ln ⁡ s ) ) = exp ⁡ ( ∫ t ~ t 0 y ( s ) s d s ) ( ln ⁡ t ln ⁡ t ~ ) ± 1 ∈ L β , $$\begin{split}\tilde{L}(t)=\exp\bigg(\int\limits_{t}^{t_{0}}\frac{y(s)}{s}ds\bigg)&=\exp\bigg(\int\limits_{\tilde{t}}^{t_{0}}\frac{y(s)}{s}ds\bigg)\exp\bigg(\pm\int\limits_{t}^{\tilde{t}}\frac{ds}{s(-\ln s)}\bigg)\\&=\exp\bigg(\int\limits_{\tilde{t}}^{t_{0}}\frac{y(s)}{s}ds\bigg)\bigg(\frac{\lnt}{\ln \tilde{t}}\bigg)^{\pm1}\in \mathfrak{L}_{\beta},\end{split}$$where β ∈ (0, 1) with E3=0 or β=1 with E3=± 1. It’s clear that (1.8) holds here. This implies that when L˜∈Lβ$ \tilde{L}\in \mathfrak{L}_{\beta} $with β ≤ 1, (1.3) is not always true if (1.17) fails.Remark 1.4In Theorem 1.3, if we substitute L˜∈Lβ0$ \tilde{L}\in \mathfrak{L}_{\beta_{0}} $(β0>β) andE3<0,β0∈(0,1),−1,β0=1,$$\begin{align}E_{3}<\begin{cases} 0,\,&\beta_{0}\in(0, 1),\\-1,\, &\beta_{0}=1,\end{cases}\end{align}$$for L˜∈Lβ$ \tilde{L}\in\mathfrak{L} _{\beta} $with (1.17), then the conclusion of Theorem 1.3 still holds.Remark 1.5L˜∈L$ \tilde{L}\in\mathcal {L} $is normalized slowly varying at zero and limt→0+tL˜′(t)L˜(t)=0$ \lim_{t\rightarrow0^{+}}\frac{t\tilde{L}'(t)}{\tilde{L}(t)}=0 $.The rest of the paper is organized as follows. In Section 2, we give some bases of Karamata regular variation theory. In Section 3, we show some auxiliary lemmas. The proofs of Theorems 1.1-1.3 are given in Section 4.2Some basic facts from Karamata regular variation theoryIn this section, we introduce some preliminaries of Karamata regular variation theory which come from [3], [32], [40]- [41].Definition 2.1A positive continuous function g defined on (0, a0], for some a0 > 0, is called regularly varying at zero with index p, denoted by g ∈ RVZp, if for each ξ>0 and some p∈R$ p \in \mathbb R $,(2.1)limt→0+g(ξt)g(t)=ξp.$$\begin{equation}\lim_{t\rightarrow 0^{+}} \frac{g(\xi t)}{g(t)}= \xi^p.\end{equation}$$In particular, when p=0, g is called slowly varying at zero.Clearly, if g ∈ RVZp, then L(t)≔ g(t)/tp is slowly varying at zero.Proposition 2.2(Uniform Convergence Theorem). If g ε RVZp, then (2.1) holds uniformlyfor ξ ε [c1, c2], where 0< c1 < c2 < a0.Proposition 2.3(Representation Theorem). A function L is slowly varying at zero if and only if it may be written in the formL ( t ) = l ( t ) e x p ∫ t a 0 y ( s ) s d s , t ≤ a 0 , $$\begin{align}L(t)=l(t) {\rm exp} \left( \int\limits_{t}^{a_{0}} \frac {y(s)}{s} ds\right), \t \leq a_0,\end{align}$$where the functions l and y are continuous and for t → 0+, y(t) → 0 and l(t) → c0 with c0 > 0.Definition 2.4The functionL ( t ) = c 0 e x p ∫ t a 0 y ( s ) s d s , t ≤ a 0 , $$L(t)=c_{0}{\rm exp} \left( \int\limits_{t}^{a_{0}} \frac {y(s)}{s} ds\right), \t \leq a_0,$$is called normalized slowly varying at zero andh(t)=tpL(t), t≥a0$$\begin{align}h(t)=t^p L(t), \t \geq a_0\end{align}$$is called normalized regularly varying at zero with index ρ (written as f ∈ NRVZp).A function h ε C1(0, a0] for some a0 > 0 belongs to NRVZp if and only iflimt→0+th′(t)h(t)=p.$$\begin{align}\lim_{t\rightarrow0^{+}}\frac{th'(t)}{h(t)}=p.\end{align}$$Proposition 2.5Let functions L, L1 be slowly varying at zero, then(i) Lp, p∈R$ p\in\mathbb{R} $, L1· L and L1 ∘ L satisfying limt→0+L(t)=0$ \lim_{t\rightarrow0^{+}}L(t)=0 $, are also slowly varying at zero;(ii) for any p > 0,tpL(t)→0 and t−pL(t)→∞ ast→0+;$$\begin{align}t^{p}L(t)\rightarrow 0 \mbox{ and }t^{-p}L(t)\rightarrow\infty\mbox{ as}t\rightarrow0^{+};\end{align}$$(iii) for any p ∈ ℝ,ln ⁡ L ( t ) / ln ⁡ t → 0 and ln ⁡ ( t p L ( t ) ) / ln ⁡ t = p as t → 0 + . $$\begin{align}\ln L(t)/\ln t\rightarrow0 \mbox{ and }\ln(t^{p}L(t))/\ln t=p \mbox{ as}\,t\rightarrow0^{+}.\end{align}$$Proposition 2.6Let g1∈(N)RVZp1 and g2∈(N)RVZp2, then g1⋅g2∈(N)RVZp1+p2.$ g_{1}\in (N)RVZ_{p_{1}}\text{ and }g_{2}\in (N)RVZ_{p_{2}}\text{, then }g_{1}\cdot g_{2}\in (N)RVZ_{p_{1}+p_{2}}. $Proposition 2.7Let g1∈(N)RVZp1,g2∈(N)RVZp2 and limt→0+g2(t)=0, then g1∘g2∈(N)RVZp1p2.$ g_{1}\in (N)RVZ_{p_{1}},\,\,g_{2}\in (N)RVZ_{p_{2}}\text{ and }\lim_{t\rightarrow0^{+}}g_{2}(t)=0\text{, then }g_{1}\circ g_{2}\in(N)RVZ_{p_{1}p_{2}}. $Proposition 2.8(Asymptotic Behavior). Let L is a slowly varying functionat zero, then for a1 > 0,(i) ∫ta1spL(s)ds∼−(1+p)−1t1+pL(t), forp<−1,t→0+;$ \int_{t}^{a_{1}}s^{p}L(s)ds\sim -(1+p)^{-1}t^{1+p}L(t),\mbox{ for} p < -1,\,t\rightarrow 0^{+}; $(ii)∫0tspL(s)ds∼(1+p)−1t1+pL(t), forp>−1,t→0+.$ \int_{0}^{t}s^{p}L(s)ds \sim (1+p)^{-1}t^{1+p}L(t), \mbox{ for}p>-1,\, t\rightarrow0^{+}. $3Auxiliary resultsIn this section, we show some useful results, which are necessary in the proofs of our results.Lemma 3.1(Lemma 2.9 in [51]). Let k ∈ Λ, thenwhen k is non-decreasing, Dk ∈ [0, 1]; and when k is non-increasing, Dk ∈ [1, ∞);limt→0+K(t)k(t)=0$ \lim_{t\rightarrow0^{+}}\frac{K(t)}{k(t)}=0 $, limt→0+K(t)tk(t)=Dk$ \lim_{t\rightarrow0^{+}}\frac{K(t)}{tk(t)}=D_{k} $and limt→0+K(t)k′(t)k2(t)=1−Dk;$ \lim_{t\rightarrow0^{+}}\frac{K(t)k'(t)}{k^{2}(t)}=1-D_{k}; $when Dk > 0 and Dk≠1, k∈NRVZ(1−Dk)/Dk;$ k\in NRVZ_{(1-D_{k})/D_{k}}; $when Dk=1, k is normalized slowlyvarying at zero.Lemma 3.2Let L˜∈Lβ$ \tilde{L}\in\mathfrak{L}_{\beta} $and (1.17) hold, then∫0tL˜(s)sds<∞;$ \int_{0}^{t}\frac{\tilde{L}(s)}{s}ds<\infty; $limt→0+(−lnt)βL˜(s)∫0tL˜(s)sds=−E3,β∈(0,1),−(E3+1),β=1;$\lim_{t\rightarrow0^{+}}(-\lnt)^{\beta}\frac{\tilde{L}(s)}{\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds}=\begin{cases}-E_{3},\, &\beta\in(0, 1),\\-(E_{3}+1),\, &\beta=1;\end{cases}$limt→0+(−lnt)βL˜′(t)tL˜(t)=E3.$\lim_{t\rightarrow0^{+}}(-\lnt)^{\beta}\frac{\tilde{L}'(t)t}{\tilde{L}(t)}= E_{3}. $Proof. (i) Bylimt→0+(−lnt)βy(t)=E3<0,β∈(0,1),−1,β=1,$$\lim\limits_{t \rightarrow 0^{+}}(-\ln t)^{\beta} y(t)=E_{3}<\left\{\begin{array}{ll}0, & \beta \in(0,1), \\ -1, & \beta=1,\end{array}\right.$$we see that for any 0<ε<−E3+τ2$0<\varepsilon<-\frac{E_{3}+\tau}{2} $withτ=0,β∈(0,1),1,β=1,$$\tau=\left\{\begin{array}{ll}0, & \beta \in(0,1), \\ 1, & \beta=1,\end{array}\right.$$there exists a small enough positive constant t* ∈ (0, 1) such thaty(t)≤E3+ε(−lnt)β,t∈(0,t∗] with E3+ε<0,β∈(0,1),−1,β=1.$$y(t)\leq \frac{E_{3}+\varepsilon}{(-\ln t)^{\beta}},\,t\in(0, t_{*}]\mbox{ with }E_{3}+\varepsilon<\begin{cases} 0,\,&\beta\in (0, 1),\\-1,\, &\beta=1.\end{cases}$$A straightforward calculation shows that for any t∈0,t⋆,$t \in\left(0, t_{\star}\right],$exp(∫tt∗y(s)sds)≤exp(∫tt∗E3+εs(−lns)βds)=exp(E3+ε1−β((−lnt)1−β−(−lnt∗)1−β)),β∈(0,1),(lntlnt∗)E3+ε,β=1.$$\begin{split}\exp\bigg(\int\limits_{t}^{t_{*}}\frac{y(s)}{s}ds\bigg)&\leq\exp\bigg(\int\limits_{t}^{t_{*}}\frac{E_{3}+\varepsilon}{s(-\ln s)^{\beta}}ds\bigg)\\&=\begin{cases}\exp\bigg(\frac{E_{3}+\varepsilon}{1-\beta}\big((-\lnt)^{1-\beta}-(-\ln t_{*})^{1-\beta}\big)\bigg),\,&\beta\in(0, 1),\\\big(\frac{\ln t}{\ln t_{*}}\big)^{E_{3}+\varepsilon},\, &\beta=1.\end{cases}\end{split}$$So, we see that there exists a large constant C > 0 such that for any t∈(0,t∗],$t\in(0, t_{*}],$(3.1)L˜(t)≤Cexp(E3+ε1−β(−lnt)1−β),β∈(0,1),C(−lnt)E3+ε,β=1.$$\begin{equation}\tilde{L}(t)\leq \begin{cases}C\exp\big(\frac{E_{3}+\varepsilon}{1-\beta}(-\lnt)^{1-\beta}\big),\,&\beta\in(0, 1),\\C(-\ln t)^{E_{3}+\varepsilon},\, &\beta=1.\end{cases}\end{equation}$$Case 1. If β∈(0, 1), it is clear that thatexp(−E3+ε1−β(−lnt)1−β)=∑n=0∞(−E3+ε1−β)n(−lnt)n(1−β)n!.$$\exp\bigg(-\frac{E_{3}+\varepsilon}{1-\beta}(-\lnt)^{1-\beta}\bigg)=\sum\limits_{n=0}^{\infty}\frac{\big(-\frac{E_{3}+\varepsilon}{1-\beta}\big)^{n}(-\lnt)^{n(1-\beta)}}{n!}.$$This together with (3.1) implies that we can choose a positive integer n* >(1-β)-1 such that(3.2)Cexp(E3+ε1−β(−lnt)1−β)=C(exp(−E3+ε1−β(−lnt)1−β))−1=C(∑n=0∞(−E3+ε1−β)n(−lnt)n(1−β)n!)−1≤C((−E3+ε1−β)n∗(−lnt)n∗(1−β))−1=C1(−lnt)−n∗(1−β),t∈(0,t∗],$$\begin{equation}\label{dll}\begin{split}C\exp\bigg(\frac{E_{3}+\varepsilon}{1-\beta}(-\lnt)^{1-\beta}\bigg)&=C\bigg(\exp\bigg(-\frac{E_{3}+\varepsilon}{1-\beta}(-\lnt)^{1-\beta}\bigg)\bigg)^{-1}\\&=C\bigg(\sum_{n=0}^{\infty}\frac{\big(-\frac{E_{3}+\varepsilon}{1-\beta}\big)^{n}(-\lnt)^{n(1-\beta)}}{n!}\bigg)^{-1}\\&\leqC\bigg(\bigg(-\frac{E_{3}+\varepsilon}{1-\beta}\bigg)^{n_{*}}(-\lnt)^{n_{*}(1-\beta)}\bigg)^{-1}\\&=C_{1}(-\ln t)^{-n_{*}(1-\beta)},\,t\in(0, t_{*}],\end{split}\end{equation}$$where C1=C(−E3+ε1−β)−n∗$C_{1}=C\big(-\frac{E_{3}+\varepsilon}{1-\beta}\big)^{-n_{*}}$. So, we have∫0t∗L˜(s)sds≤C∫0t∗exp(E3+ε1−β(−lns)1−β)sds≤C1∫0t∗dss(−lns)n∗(1−β)=C1(−lnt∗)1−n∗(1−β)n∗(1−β)−1<∞.$$\begin{split}\int\limits_{0}^{t_{*}}\frac{\tilde{L}(s)}{s}ds &\leqC\int\limits_{0}^{t_{*}}\frac{\exp\big(\frac{E_{3}+\varepsilon}{1-\beta}(-\ln s)^{1-\beta}\big)}{s}ds\\&\leq C_{1}\int\limits_{0}^{t_{*}}\frac{ds}{s (-\ln s)^{n_{*}(1-\beta)}}=\frac{C_{1}(-\lnt_{*})^{1-n_{*}(1-\beta)}}{n_{*}(1-\beta)-1}<\infty.\end{split}$$Case 2. If β=1, then by (3.1) we obtain∫0t∗L˜(s)sds≤∫0t∗C(−lns)E3+εsds=−C(−lnt∗)1+E3+ε1+E3+ε<∞.$$\int\limits_{0}^{t_{*}}\frac{\tilde{L}(s)}{s}ds\leq\int\limits_{0}^{t_{*}}\frac{C(-\lns)^{E_{3}+\varepsilon}}{s}ds=-\frac{C(-\lnt_{*})^{1+E_{3}+\varepsilon}}{1+E_{3}+\varepsilon}<\infty.$$Combining Cases 1-2, we obtain (i) holds.(ii) It follows by (3.1)-(3.2) that limt→0+(−lnt)βL˜(t)=0.$\lim_{t\rightarrow0^{+}}(-\ln t)^{\beta}\tilde{L}(t)=0.$By the l'Hospital’s rule, we obtainlimt→0+(−lnt)βL˜(s)∫0tL˜(s)sds=limt→0+(−(−lnt)βy(t)−β(−lnt)β−1)=−E3,β∈(0,1),−(E3+1),β=1.$$\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}\frac{\tilde{L}(s)}{\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds}=\lim\limits_{t\rightarrow0^{+}}\big(-(-\lnt)^{\beta}y(t)-\beta(-\ln t)^{\beta-1}\big)=\begin{cases}-E_{3},\, &\beta\in(0, 1),\\-(E_{3}+1),\, &\beta=1.\end{cases}$$(iii) A direct calculation shows thatlimt→0+(−lnt)βL˜′(t)tL˜(t)=limt→0+(−lnt)βy(t)=E3.$$\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}\frac{\tilde{L}'(t)t}{\tilde{L}(t)}=\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}y(t)=E_{3}.$$□Define(3.3)φ(t):=∫0t((N+1)G(s))−1/(N+1)ds,t>0,$$\begin{equation}\varphi(t):=\int\limits_{0}^{t}((N+1)G(s))^{-1/(N+1)}ds,\,t>0,\end{equation}$$where φ is the inverse of solution ψ to (1.4).Lemma 3.3Let g satisfy (g1)-(g2), thenlimt→0+tg′(t)g(t)=−γ;$\lim_{t\rightarrow0^{+}}\frac{tg'(t)}{g(t)}=-\gamma;$G(t)∼t1−γγ−1Lˆ0(t),t→0+;$G(t)\sim\frac{t^{1-\gamma}}{\gamma-1}\hat{L}_{0}(t),\,t\rightarrow0^{+};$φ(t)∼(γ−1N+1)1/(N+1)N+1γ+Ntγ+NN+1(Lˆ0(t))−1/(N+1),t→0+,$\varphi(t)\sim\big(\frac{\gamma-1}{N+1}\big)^{1/(N+1)}\frac{N+1}{\gamma+N}t^{\frac{\gamma+N}{N+1}}(\hat{L}_{0}(t))^{-1/(N+1)},\,t\rightarrow0^{+},$ whereLˆ0(t)=c0exp(∫ta1f(s)sds),c0=g(a1)a1γ;$$\hat{L}_{0}(t)=c_{0}\exp\bigg(\int\limits_{t}^{a_{1}}\frac{f(s)}{s}ds\bigg),\,c_{0}=g(a_{1})a_{1}^{\gamma};$$limt→0+φ(t)tφ′(t)=limt→0+φ(t)((N+1)G(t))1/(N+1)t=N+1γ+N,i.e., φ∈NRVZγ+NN+1;$\lim_{t\rightarrow0^{+}}\frac{\varphi(t)}{t\varphi'(t)}=\lim_{t\rightarrow0^{+}}\frac{\varphi(t)((N+1)G(t))^{1/(N+1)}}{t}=\frac{N+1}{\gamma+N},\mbox{i.e., }\varphi\inNRVZ_{\frac{\gamma+N}{N+1}};$limt→0+((N+1)G(t))N/(N+1)g(t)φ(t)=γ+Nγ−1.$\lim_{t\rightarrow0^{+}}\frac{((N+1)G(t))^{N/(N+1)}}{g(t)\varphi(t)}=\frac{\gamma+N}{\gamma-1}.$proof. (i) It follows by (g2) that (i) holds.(ii)-(iii) By (g2) we see that(3.4)g(t)=t−γLˆ0(t),t∈(0,a1].$$\begin{equation}g(t)=t^{-\gamma}\hat{L}_{0}(t),\, t\in(0, a_{1}].\end{equation}$$It follows by Proposition 2.8 (i) that (ii) holds. Furthermore, we have((N+1)G(t))−1/(N+1)∼(γ−1N+1)1/(N+1)t(γ−1)/(N+1)(Lˆ0(t))−1/(N+1),t→0+.$$((N+1)G(t))^{-1/(N+1)}\sim\bigg(\frac{\gamma-1}{N+1}\bigg)^{1/(N+1)}t^{(\gamma-1)/(N+1)}(\hat{L}_{0}(t))^{-1/(N+1)},\,t\rightarrow0^{+}.$$It follows by Proposition 2.8 (ii) that (iii) holds.(iv)-(v) (3.3) implies that(3.5)φ′(t)=((N+1)G(t))−1/(N+1),t>0.$$\begin{equation}\varphi'(t)=((N+1)G(t))^{-1/(N+1)},\,t>0.\end{equation}$$By (3.4)-(3.5) and (i)-(ii), we obtain (iv)-(v) hold.Lemma 3.4Suppose that g satisfies (g1)-(g2). In particular, if (S2) holds in (g2), we suppose that (g3) holds. And if θ = 0 in (g3), we further suppose that (g4) holds. Then,limt→0+υ(t)(tg′(t)g(t)+γ)=χ1,$$\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=\chi_{1},$$whereχ1=−σ,if(S2)and(g3)−(g4)holdwithθ=0in(g3),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ>0),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ(N+1)>(γ+N)ρ),υ(t)=(φ(t))−ρwithρ∈(0,1];$$\chi_{1}=\begin{cases}-\sigma, &\,{ if }\mathbf{(S_{2})} \,{ and } \,\mathbf{(g_{3})}\,{-}\mathbf{(g_{4})} \,{ hold\, with }\, \theta=0 \,{ in }\,\mathbf{(g_{3})}, \,\upsilon(t)=(-\ln t)^{\beta},\\0, &\,{ if }\,\mathbf{(S_{1})} \,{ holds\, (or}\,\mathbf{(S_{2})}\,{ and }\, \mathbf{(g_{3})} \,{ hold\, with }\,\theta>0),\, \upsilon(t)=(-\ln t)^{\beta},\\0, &\,{ if }\mathbf{(S_{1})} \,{ holds\, (or}\,\mathbf{(S_{2})}\,{ and } \mathbf{(g_{3})} \,{ hold\, with }\,\theta(N+1)>(\gamma+N)\rho),\\&\upsilon(t)=(\varphi(t))^{-\rho}\,{ with }\,\rho\in(0,1];\\\end{cases}$$limt→0+υ(t)(G(t)tg(t)−1γ−1)=χ2,$$\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)=\chi_{2},$$whereχ2=−σ(γ−1)2,if(S2)and(g3)−(g4)holdwithθ=0in(g3),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ>0),υ(t)=(−lnt)β,0,if(S1)holdsandγ>(1+ρ)N+1N+1−ρin(g2),υ(t)=(φ(t))−ρ,ρ∈(0,1],0,if(S2)and(g3)holdwithθ(N+1)ρ−N>γ>(1+ρ)N+1N+1−ρ,υ(t)=(φ(t))−ρwithρ∈(0,1];$$\chi_{2}=\begin{cases}-\frac{\sigma}{(\gamma-1)^{2}}, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\, \mathbf{(g_{3})}\,{-}\mathbf{(g_{4})} \,{ hold\, with }\, \theta=0 \,{ in }\,\mathbf{(g_{3})},\,\upsilon(t)=(-\ln t)^{\beta},\\0, &\,{ if }\mathbf{(S_{1})} \,{ holds\, (or\,}\mathbf{(S_{2})}\,{ and }\, \mathbf{(g_{3})} \,{ hold \,with}\,\theta>0), \,\upsilon(t)=(-\lnt)^{\beta},\\0, &\,{ if }\,\mathbf{(S_{1})}\,{ holds\, and }\,\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \,{ in }\,\mathbf{(g_{2})},\,\upsilon(t)=(\varphi(t))^{-\rho},\, \rho\in(0, 1],\\0, &\,{ if }\,\mathbf{(S_{2})}\,{ and }\, \mathbf{(g_{3})} \,{hold\, with}\,\frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho},\\&\upsilon(t)=(\varphi(t))^{-\rho}\,{ with }\, \rho\in(0, 1];\end{cases}$$limt→0+υ(t)(((N+1)G(t))N/(N+1)g(t)φ(t)−γ+Nγ−1)=χ3,$$\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{((N+1)G(t))^{N/(N+1)}}{g(t)\varphi(t)}-\frac{\gamma+N}{\gamma-1}\bigg)=\chi_{3},$$whereχ3=−(N+1)σ(γ−1)2,if(S2)and(g3)−(g4)holdwithθ=0in(g3),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ>0),υ(t)=(−lnt)β,0,if(S1)holdsandγ>(1+ρ)N+1N+1−ρin(g2),υ(t)=(φ(t))−ρ,ρ∈(0,1],0,if(S2)and(g3)holdwithθ(N+1)ρ−N>γ>(1+ρ)N+1N+1−ρ,υ(t)=(φ(t))−ρwithρ∈(0,1];$$\chi_{3}=\begin{cases}-\frac{(N+1)\sigma}{(\gamma-1)^{2}}, &\,{ if }\,\mathbf{(S_{2})} \,{ and } \,\mathbf{(g_{3})}\,{-}\,\mathbf{(g_{4})} \,{ hold \,with }\, \theta=0 \,{ in }\,\mathbf{(g_{3})}, \,\upsilon(t)=(-\ln t)^{\beta},\\0, &\,{ if }\,\mathbf{(S_{1})} \,{ holds\, (or}\,\mathbf{(S_{2})}\,{ and } \,\mathbf{(g_{3})} \,{ hold\, with}\,\theta>0), \,\upsilon(t)=(-\lnt)^{\beta},\\0, &\,{ if }\,\mathbf{(S_{1})}\,{ holds\, and}\,\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \,{ in }\,\mathbf{(g_{2})}, \,\upsilon(t)=(\varphi(t))^{-\rho},\,\rho\in(0, 1],\\0, &\,{ if }\,\mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})} \,{ hold \,with }\,\frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho},\\&\upsilon(t)=(\varphi(t))^{-\rho}\,{ with }\,\rho\in(0, 1];\end{cases}$$let ξ be a positive constant, thenlimt→0+υ(t)g(ξt)ξNg(t)−ξ−(γ+N)=χ4,$$\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\left(\frac{g(\xi t)}{\xi^{N}g(t)}-\xi^{-(\gamma+N)}\right)=\chi_{4},$$whereχ4=−σlnξξγ+N,ifS2andg3−g4holdwithθ=0ing3,v(t)=(−lnt)β,0,ifS1holdsorS2andg3holdwithθ>0,v(t)=(−lnt)β,0,ifS1holdsorS2andg3holdwithθ(N+1)>(γ+N)ρ,v(t)=(φ(t))−ρwithρ∈(0,1].$$\chi_{4}=\,\left\{\begin{array}{ll}-\frac{\sigma \ln \xi}{\xi^{\gamma+N}}, & \, { if }\,\left(\mathbf{S}_{\mathbf{2}}\right) \, { and }\,\left(\mathbf{g}_{3}\right)-\,\left(\mathbf{g}_{4}\right) \, { hold\, with } \,\theta=0 \, { in }\,\left(\mathbf{g}_{3}\right), v(t)=(-\ln t)^{\beta}, \\ 0, & \, { if }\,\left(\mathbf{S}_{1}\right) \, { holds }\,\left(\, { or }\,\left(\mathbf{S}_{\mathbf{2}}\right) \, { and }\,\left(\mathbf{g}_{3}\right) \, { hold\, with }\, \theta>0\right), v(t)=(-\ln t)^{\beta}, \\ 0, & \, { if }\,\left(\mathbf{S}_{1}\right) \, { holds }\,\left(\, { or }\,\left(\mathbf{S}_{2}\right) \, { and }\,\left(\mathbf{g}_{3}\right) \, { hold\, with }\, \theta(N+1)>(\gamma+N) \rho\right), \\ & v(t)=(\varphi(t))^{-\rho} \, { with }\, \rho \in(0,1] .\end{array}\right.$$Proof. (i) When (S2) and (g3)-(g4) hold with θ = 0 in (g3), we have(3.6)limt→0+(−lnt)β(tg′(t)g(t)+γ)=−limt→0+(−lnt)βf(t)=−σ.$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=-\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}f(t)=-\sigma.\end{equation}$$When (S1) holds, we havetg′(t)g(t)=−γ,t∈(0,a1].$$\frac{tg'(t)}{g(t)}=-\gamma,\,t\in(0, a_{1}].$$This implies that for υ (t)=(-ln t)β orυ (t)=(φ (t))-ρ, we obtain(3.7)limt→0+υ(t)(tg′(t)g(t)+γ)=0.$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=0.\end{equation}$$When (S2) and (g3)-(g4) hold with θ > 0, we conclude by Proposition 2.5 (ii) that(3.8)limt→0+(−lnt)β(tg′(t)g(t)+γ)=−limt→0+(−lnt)βf(t)=−limt→0+tθ(−lnt)βL(t)=0,$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=-\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}f(t)=-\lim\limits_{t\rightarrow0^{+}}t^{\theta}(-\lnt)^{\beta}L(t)=0,\end{equation}$$where L ∈ NRVZ0. Moreover, it follows by Lemma (3.3) (iv) and Proposition (2.7) that(3.9)1/φρ∈NRVZ−(γ+N)ρN+1.$$\begin{equation}1/\varphi^{\rho}\in NRVZ_{-\frac{(\gamma+N)\rho}{N+1}}.\end{equation}$$When (g3) holds with θ (N + 1) > (γ + N)ρ, we conclude by Proposition 2.6 and Proposition 2.5 (ii) that(3.10)limt→0+(φ(t))−ρ(tg′(t)g(t)+γ)=−limt→0+(φ(t))−ρf(t)=−limt→0+tθ−(γ+N)ρN+1L(t)=0,$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}(\varphi(t))^{-\rho}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=-\lim\limits_{t\rightarrow0^{+}}(\varphi(t))^{-\rho}f(t)=-\lim\limits_{t\rightarrow0^{+}}t^{\theta-\frac{(\gamma+N)\rho}{N+1}}L(t)=0,\end{equation}$$where L ∈ NRVZ0. So, (i) follows by (3.6)-(3.10).(ii) By (g2), we have−tg′(t)=γg(t)+g(t)f(t),t∈(0,a1].$$-tg'(t)=\gamma g(t)+g(t)f(t),\,t\in(0, a_{1}].$$Integrating it from t to a1 and integration by parts, we obtaintg(t)=(γ−1)G(t)+∫ta1g(s)f(s)ds+c,t∈(0,a1],$$tg(t)=(\gamma-1)G(t)+\int\limits_{t}^{a_{1}}g(s)f(s)ds+c,\, t\in(0, a_{1}],$$i.e.,(3.11)G(t)tg(t)−1γ−1=−1γ−1∫ta1g(s)f(s)dstg(t)−c(γ−1)g(t)t,t∈(0,a1],$$\begin{equation}\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}=-\frac{1}{\gamma-1}\frac{\int_{t}^{a_{1}}g(s)f(s)ds}{tg(t)}-\frac{c}{(\gamma-1)g(t)t},\,t\in(0,a_{1}],\end{equation}$$where c is a constant. The condition (g2) implies that g ∈ NRVZ-γ with γ > 1. Moreover, by t ↦ (-ln t)β ∈ NRVZ0 and (3.9), we see thatυ∈NRVZ0, if υ(t)=(−lnt)β,NRVZ−(γ+N)ρN+1, if υ(t)=(φ(t))−ρ.$$\upsilon\in\begin{cases}NRVZ_{0},\, &\mbox{ if }\upsilon(t)=(-\ln t)^{\beta},\\NRVZ_{-\frac{(\gamma+N)\rho}{N+1}},\,&\mbox{ if }\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$We conclude by Proposition 2.7 and Proposition 2.6 thatt↦tg(t)(υ(t))−1∈NRVZ1−γ, if υ(t)=(−lnt)β,NRVZ(1−γ)(N+1)+(γ+N)ρN+1, if υ(t)=(φ(t))−ρ.$$t\mapsto tg(t)(\upsilon(t))^{-1}\in\begin{cases}NRVZ_{1-\gamma},\, &\mbox{ if }\upsilon(t)=(-\ln t)^{\beta},\\NRVZ_{\frac{(1-\gamma)(N+1)+(\gamma+N)\rho}{N+1}},\,&\mbox{ if}\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$So, we can take L ∈ NRVZ0 such thattg(t)(υ(t))−1=tςL(t)$$tg(t)(\upsilon(t))^{-1}=t^{\varsigma}L(t)$$withς=1−γ, if υ(t)=(−lnt)β,(1−γ)(N+1)+(γ+N)ρN+1, if υ(t)=(φ(t))−ρ.$$\varsigma=\begin{cases}1-\gamma,\, &\mbox{ if }\upsilon(t)=(-\ln t)^{\beta},\\\frac{(1-\gamma)(N+1)+(\gamma+N)\rho}{N+1},\,&\mbox{ if}\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$It’s clear thatς<0, if γ>1 in (g2) and υ(t)=(−lnt)β,ς<0, if γ>(1+ρ)N+1N+1−ρ in (g2) and υ(t)=(φ(t))−ρ.$$\begin{cases}\varsigma<0,\, &\mbox{ if }\gamma>1 \mbox{ in }\mathbf{(g_{2})} \mbox{ and }\upsilon(t)=(-\ln t)^{\beta},\\\varsigma<0,\,&\mbox{ if }\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \mbox{in }\mathbf{(g_{2})}\mbox{ and }\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$By Proposition 2.5 (ii), we havelimt→0+tg(t)(υ(t))−1=∞,$$\lim\limits_{t\rightarrow0^{+}}tg(t)(\upsilon(t))^{-1}=\infty,$$i.e.,(3.12)limt→0+−c(γ−1)tg(t)(υ(t))−1=0.$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}-\frac{c}{(\gamma-1)tg(t)(\upsilon(t))^{-1}}=0.\end{equation}$$Combining with (3.6)-(3.10), by the l'Hospital’s rule, we can obtain(3.13)limt→0+υ(t)−∫ta1g(s)f(s)ds(γ−1)tg(t)=limt→0+∫a1tg(s)f(s)ds(γ−1)tg(t)(υ(t))−1=limt→0+1γ−1g(t)f(t)g(t)(υ(t))−1+tg′(t)(υ(t))−1+tg(t)[(υ(t))−1]′=limt→0+1γ−1υ(t)f(t)1+tg′(t)g(t)+t[(υ(t))−1]′(υ(t))−1=limt→0+(−lns)βf(s)−(γ−1)2, if γ>1 in (g2) and υ(t)=(−lnt)βlimt→0+(N+1)(φ(t))−ρf(t)(γ−1)((1−γ)(N+1)+(γ+N)ρ), if γ>(1+ρ)N+1N+1−ρ in (g2) and υ(t)=(φ(t))−ρ=χ2.$$\begin{equation}\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\frac{-\int_{t}^{a_{1}}g(s)f(s)ds}{(\gamma-1)tg(t)}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\int_{a_{1}}^{t}g(s)f(s)ds}{(\gamma-1)tg(t)(\upsilon(t))^{-1}}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{1}{\gamma-1}\frac{g(t)f(t)}{g(t)(\upsilon(t))^{-1}+tg'(t)(\upsilon(t))^{-1}+tg(t)[(\upsilon(t))^{-1}]'}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{1}{\gamma-1}\frac{\upsilon(t)f(t)}{1+\frac{tg'(t)}{g(t)}+\frac{t[(\upsilon(t))^{-1}]'}{(\upsilon(t))^{-1}}}\\=&\begin{cases}\lim\limits_{t\rightarrow0^{+}}\frac{(-\ln s)^{\beta}f(s)}{-(\gamma-1)^{2}},\, &\mbox{ if }\gamma>1 \mbox{ in }\mathbf{(g_{2})} \mbox{ and }\upsilon(t)=(-\ln t)^{\beta}\\\lim\limits_{t\rightarrow0^{+}}\frac{(N+1)(\varphi(t))^{-\rho}f(t)}{(\gamma-1)((1-\gamma)(N+1)+(\gamma+N)\rho)},\,&\mbox{if }\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \mbox{ in}\mathbf{(g_{2})}\mbox{ and }\upsilon(t)=(\varphi(t))^{-\rho}\end{cases}\\=&\chi_{2}.\end{split}\end{equation}$$(3.11)-(3.13) imply thatlimt→0+υ(t)G(t)tg(t)−1γ−1=−limt→0+∫ta1g(s)f(s)ds(γ−1)tg(t)(υ(t))−1−limt→0+c(γ−1)tg(t)(υ(t))−1=χ2.$$\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\left(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\right)\\=-&\lim\limits_{t\rightarrow0^{+}}\frac{\int_{t}^{a_{1}}g(s)f(s)ds}{(\gamma-1)tg(t)(\upsilon(t))^{-1}}-\lim\limits_{t\rightarrow0^{+}}\frac{c}{(\gamma-1)tg(t)(\upsilon(t))^{-1}}=\chi_{2}.\\\end{split}$$(iii) By Lemma 3.3 (iv), Proposition 2.7 and Proposition 2.6, we havet↦(υ(t))−1φ(t)∈NRVZ(γ+N)ρN+1, if υ(t)=(−lnt)β,NRVZ(γ+N)(ρ+1)N+1, if υ(t)=(φ(t))−ρ.$$t\mapsto(\upsilon(t))^{-1}\varphi(t)\in\begin{cases}NRVZ_{\frac{(\gamma+N)\rho}{N+1}},\, &\mbox{ if } \upsilon(t)=(-\ln t)^{\beta},\\NRVZ_{\frac{(\gamma+N)(\rho+1)}{N+1}},\,&\mbox{ if}\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$This implies that(υ(t))−1φ(t)→0 as t→0+.$$(\upsilon(t))^{-1}\varphi(t)\rightarrow0 \mbox{as }t\rightarrow0^{+}.$$Moreover, it follows by Lemma 3.3 (v) that((N+1)G(t))1/(N+1)g(t)−γ+Nγ−1φ(t)→0 as t→0+.$$\frac{((N+1)G(t))^{1/(N+1)}}{g(t)}-\frac{\gamma+N}{\gamma-1}\varphi(t)\rightarrow0\mbox{ as }t\rightarrow0^{+}.$$If ν(t) = (− ln t)β, then we have by the l’Hospital’s rule, (3.5) and (i)-(ii) that(3.14)limt→0+υ(t)(((N+1)G(t))N/(N+1)g(t)φ(t)−γ+Nγ−1)=limt→0+((N+1)G(t))N/(N+1)g(t)−γ+Nγ−1φ(t)(−lnt)−βφ(t)=limt→0+−N((N+1)G(t))−1/(N+1)g2(t)−((N+1)G(t))N/(N+1)g′(t)g2(t)−γ+Nγ−1φ′(t)(−lnt)−βφ′(t)+β(−lnt)−β−1φ(t)t−1=limt→0+−(N+1)(−lnt)β(G(t)tg(t)tg′(t)g(t)+γγ−1)=limt→0+−(N+1)(−lnt)β[(G(t)tg(t)−1γ−1)(tg′(t)g(t)+γ)−γ(G(t)tg(t)−1γ−1)+1γ−1(tg′(t)g(t)+γ)]=−(N+1)σ(γ−1)2, if (S2) and (g3)-(g4) hold with θ=0 in (g3),0, if (S1) holds (or (S2) and (g3) hold with θ>0).$$\begin{equation}\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{((N+1)G(t))^{N/(N+1)}}{g(t)\varphi(t)}-\frac{\gamma+N}{\gamma-1}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\frac{((N+1)G(t))^{N/(N+1)}}{g(t)}-\frac{\gamma+N}{\gamma-1}\varphi(t)}{(-\lnt)^{-\beta}\varphi(t)}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\frac{-N((N+1)G(t))^{-1/(N+1)}g^{2}(t)-((N+1)G(t))^{N/(N+1)}g'(t)}{g^{2}(t)}-\frac{\gamma+N}{\gamma-1}\varphi'(t)}{(-\ln t)^{-\beta}\varphi'(t)+\beta(-\ln t)^{-\beta-1}\varphi(t)t^{-1}}\\=&\lim\limits_{t\rightarrow0^{+}}-(N+1)(-\lnt)^{\beta}\bigg(\frac{G(t)}{tg(t)}\frac{tg'(t)}{g(t)}+\frac{\gamma}{\gamma-1}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}-(N+1)(-\ln t)^{\beta}\bigg[\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)-\gamma\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)\\&+\frac{1}{\gamma-1}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)\bigg]=\begin{cases}-\frac{(N+1)\sigma}{(\gamma-1)^{2}},\, &\mbox{ if }\mathbf{(S_{2})}\mbox{ and } \mathbf{(g_{3})}\mbox{-}\mathbf{(g_{4})} \mbox{ hold with } \theta=0 \mbox{ in }\mathbf{(g_{3})},\\0,\, &\mbox{ if }\mathbf{(S_{1})} \mbox{ holds (or } \mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with } \theta>0).\\\end{cases}\end{split}\end{equation}$$If ν(t) = (φ(t))−ρ, then we have by the l'Hospital’s rule, (3.5) and (i)-(ii) that(3.15)limt→0+υ(t)(((N+1)G(t))N/(N+1)g(t)φ(t)−γ+Nγ−1)=limt→0+((N+1)G(t))N/(N+1)g(t)−γ+Nγ−1φ(t)(φ(t))ρ+1=limt→0+−N((N+1)G(t))−1/(N+1)g2(t)−((N+1)G(t))N/(N+1)g′(t)g2(t)−γ+Nγ−1φ′(t)(ρ+1)(φ(t))ρφ′(t)=limt→0+−N+1ρ+1(φ(t))−ρ(G(t)tg(t)tg′(t)g(t)+γγ−1)=limt→0+N+1ρ+1(φ(t))−ρ[−(G(t)tg(t)−1γ−1)(tg′(t)g(t)+γ)+γ(G(t)tg(t)−1γ−1)−1γ−1(tg′(t)g(t)+γ)]=0, if (S1) holds and γ>(1+ρ)N+1N+1−ρ in (g2),0, if (S2) and (g3) hold with θ(N+1)ρ−N>γ>(1+ρ)N+1N+1−ρ.$$\begin{equation}\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{((N+1)G(t))^{N/(N+1)}}{g(t)\varphi(t)}-\frac{\gamma+N}{\gamma-1}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\frac{((N+1)G(t))^{N/(N+1)}}{g(t)}-\frac{\gamma+N}{\gamma-1}\varphi(t)}{(\varphi(t))^{\rho+1}}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\frac{-N((N+1)G(t))^{-1/(N+1)}g^{2}(t)-((N+1)G(t))^{N/(N+1)}g'(t)}{g^{2}(t)}-\frac{\gamma+N}{\gamma-1}\varphi'(t)}{(\rho+1)(\varphi(t))^{\rho}\varphi'(t)}\\=&\lim\limits_{t\rightarrow0^{+}}-\frac{N+1}{\rho+1}(\varphi(t))^{-\rho}\bigg(\frac{G(t)}{tg(t)}\frac{tg'(t)}{g(t)}+\frac{\gamma}{\gamma-1}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}\frac{N+1}{\rho+1}(\varphi(t))^{-\rho}\bigg[-\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)+\gamma\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)\\&-\frac{1}{\gamma-1}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)\bigg]=\begin{cases} 0,\,&\mbox{ if }\mathbf{(S_{1}) } \mbox{ holds and }\gamma>\frac{(1+\rho)N+1}{N+1-\rho}\mbox{ in }\mathbf{(g_{2})},\\0,\,&\mbox{ if }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{hold with}\frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho}.\end{cases}\end{split}\end{equation}$$(3.14)-(3.15) imply that (iii) holds.(iv) When (S1) holds in (g2), we haveg(ξt)ξNg(t)=ξ−(γ+N),t∈(0,a1].$$\frac{g(\xi t)}{\xi^{N}g(t)}=\xi^{-(\gamma+N)},\,t\in(0, a_{1}].$$In this case, for υ(t) = (−ln t)β or υ(t) = (φ(t))−ρ, we obtain(3.16)limt→0+υ(t)(g(ξt)ξNg(t)−ξ−(γ+N))=0.$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{g(\xit)}{\xi^{N}g(t)}-\xi^{-(\gamma+N)}\bigg)=0.\end{equation}$$Now, we investigate the case that (S2) holds in (g2). If ξ = 1, then the result is obvious. If ξ ≠ 1, then we haveg(ξt)ξNg(t)−ξ−(γ+N)=ξ−(γ+N)(exp(∫ξttf(s)sds)−1).$$\frac{g(\xi t)}{\xi^{N}g(t)}-\xi^{-(\gamma+N)}=\xi^{-(\gamma+N)}\bigg(\exp\bigg(\int\limits_{\xit}^{t}\frac{f(s)}{s}ds\bigg)-1\bigg).$$It follows by Proposition 2.2 thatlimt→0+f(st)s=0andlimt→0+f(st)sf(t)=sθ−1$$\lim\limits_{t\rightarrow0^{+}}\frac{f(st)}{s}=0\, \mbox{and}\lim\limits_{t\rightarrow0^{+}}\frac{f(st)}{sf(t)}=s^{\theta-1}$$uniformly with respect to s ∈ [1, ξ] or s ∈ [ξ, 1]. So, we havelim t → 0 + ∫ ξ t t f ( τ ) τ d τ τ := t s _ _ lim t → 0 + ∫ ξ 1 f ( t s ) s d s = 0 $$ \lim\limits_{t\rightarrow0^{+}}\int\limits_{\xi t}^{t}\frac{f(\tau)}{\tau}d\tau^\underline{\underline{\tau:=ts}}\lim_{t\rightarrow0^{+}}\int\limits_{\xi}^{1}\frac{f(ts)}{s}ds=0 $$andlimt→0+∫ξ1f(st)sf(t)ds=∫ξ1sθ−1ds=−lnξ, if g3 holds with θ=0,1−ξθθ, if g3 holds with θ>0.$$\lim\limits_{t \rightarrow 0^{+}} \int\limits_{\xi}^{1} \frac{f(s t)}{s f(t)} d s=\int\limits_{\xi}^{1} s^{\theta-1} d s=\left\{\begin{array}{ll}-\ln \xi, & \text { if }\left(\mathbf{g}_{3}\right) \text { holds with } \theta=0, \\ \frac{1-\xi^{\theta}}{\theta}, & \text { if }\left(\mathbf{g}_{3}\right) \text { holds with } \theta>0.\end{array}\right.$$Since er−1 ∼ r as r → 0+, we conclude by (3.6), (3.8)-(3.10) thatlimt→0+υ(t)(g(ξt)ξNg(t)−ξ−(γ+N))=limt→0+ξ−(γ+N)υ(t)f(t)∫ξ1f(st)sf(t)ds=−σlnξξγ+N, if (g3)-(g4) hold with θ=0 in (g3),υ(t)=(−lnt)β,0, if (g3) hold with θ>0,υ(t)=(−lnt)β,0, if (g3) hold with θ(N+1)>(γ+N)ρ,υ(t)=(φ(t))−ρ.$$\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{g(\xi t)}{\xi^{N}g(t)}-\xi^{-(\gamma+N)}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}\xi^{-(\gamma+N)}\upsilon(t)f(t)\int\limits_{\xi}^{1}\frac{f(st)}{sf(t)}ds\\=&\begin{cases}-\frac{\sigma\ln\xi}{\xi^{\gamma+N}}, &\mbox{ if }\mathbf{(g_{3})}\mbox{-}\mathbf{(g_{4})} \mbox{ hold with } \theta=0 \mbox{ in }\mathbf{(g_{3})},\, \upsilon(t)=(-\ln t)^{\beta},\\0, &\mbox{ if }\mathbf{(g_{3})} \mbox{ hold with }\theta>0,\, \upsilon(t)=(-\ln t)^{\beta},\\0, &\mbox{ if }\mathbf{(g_{3})} \mbox{ hold with}\theta(N+1)>(\gamma+N)\rho,\,\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}\end{split}$$This, combined with (3.16), shows that (iv) holds. □Lemma 3.5(Lemma 2.3 in [48]) Let L˜∈L$\tilde{L}\in\mathcal {L}$, thenlimt→0+L˜(t)∫tt0L˜(s)sds=0.$$\lim\limits_{t\rightarrow0^{+}}\frac{\tilde{L}(t)}{\int_{t}^{t_{0}}\frac{\tilde{L}(s)}{s}ds}=0.$$If further ∫0t0L˜(s)sds<∞$ \int_{0}^{t_{0}}\frac{\tilde{L}(s)}{s}ds<\infty $, thenlimt→0+L˜(t)∫0tL˜(s)sds=0.$$\lim\limits_{t\rightarrow0^{+}}\frac{\tilde{L}(t)}{\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds}=0.$$Lemma 3.6Suppose that g satisfies (g1)-(g2). In particular, if (S2) holds in (g2), we suppose that (g3) holds, and if θ = 0 in (g3), we further suppose that (g4) holds. ψ isuniquely determined by (1.4). Then,ψ′(s)=((N+1)G(ψ(t)))1N+1,ψ(t)>0,ψ(0)=0$ \,\psi'(s)=((N+1)G(\psi(t)))^{\frac{1}{N+1}},\, \,\psi(t)>0,\, \psi(0)=0$and−ψ′′(t)=((N+1)G(ψ(t)))1−NN+1g(ψ(t)),t∈(0,a1);$$-\psi''(t)=((N+1)G(\psi(t)))^{\frac{1-N}{N+1}}g(\psi(t)),\, t\in(0, a_{1});$$−(ψ′(t))N−1ψ′′(t)=g(ψ(t));$-(\psi'(t))^{N-1}\psi''(t)=g(\psi(t));$limt→0+tψ′(t)ψ(t)=N+1γ+N,i.e.,ψ∈NRVZN+1γ+N;$\lim\limits_{t\rightarrow0^{+}}\frac{t\psi'(t)}{\psi(t)}=\frac{N+1}{\gamma+N},\, i.e.,\, \psi\in NRVZ_{\frac{N+1}{\gamma+N}};$limt→0+ψ′(t)tψ′′(t)=−γ+Nγ−1,i.e.,ψ′∈NRVZ−γ−1γ+N;$\lim\limits_{t\rightarrow0^{+}}\frac{\psi'(t)}{t\psi''(t)}=-\frac{\gamma+N}{\gamma-1},\, i.e.,\, \psi'\in NRVZ_{-\frac{\gamma-1}{\gamma+N}};$if k∈Λ,thenlimt→0+lntlnψ(K(t))=(γ+N)DkN+1;$k\in\Lambda, \,then\,\lim\limits_{t\rightarrow0^{+}}\frac{\ln t}{\ln\psi(K(t))}=\frac{(\gamma+N)D_{k}}{N+1};$if L˜∈Land(1.3)holds,thenlimt→0+lnψ((∫0tL˜(s)sds)NN+1)lnt=0$\tilde{L}\in\mathcal {L}\,and\, (1.3)\, holds,\, then\,\lim\limits_{t\rightarrow0^{+}}\frac{\ln\psi\big(\big(\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds\big)^{\frac{N}{N+1}}\big)}{\lnt}=0$;limt→0+(−lnψ(t))β(ψ′(t)tψ′′(t)+γ+Nγ−1)=(N+1)σ(γ−1)2,if(S2)and(g3)−(g4)holdwithθ=0in(g3),0,if(S1)holds(or(S2)and(g3)holdwithθ>0);$$\begin{split}&\lim\limits_{t\rightarrow0^{+}}(-\ln\psi(t))^{\beta}\bigg(\frac{\psi'(t)}{t\psi''(t)}+\frac{\gamma+N}{\gamma-1}\bigg)\\&=\begin{cases}\frac{(N+1)\sigma}{(\gamma-1)^{2}}, &\,{ if }\, \mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})}\,{-}\mathbf{(g_{4})} \,{ hold\, with }\,\theta=0 \,{ in }\,\mathbf{(g_{3})},\\0, &\,{ if }\,\mathbf{(S_{1})} \,{ holds\, (or }\,\mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})} \,{ hold \,with }\, \theta>0);\end{cases}\end{split}$$let ρ ∈ (0, 1], thenlim t → 0 + t − ρ ( ψ ′ ( t ) t ψ ″ ( t ) + γ + N γ − 1 ) = 0 , i f ( S 1 ) h o l d s a n d γ > ( 1 + ρ ) N + 1 N + 1 − ρ i n ( g 2 ) , 0 , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ ( N + 1 ) ρ − N > γ > ( 1 + ρ ) N + 1 N + 1 − ρ ; $$\begin{split}&\lim_{t\rightarrow0^{+}}t^{-\rho}\bigg(\frac{\psi'(t)}{t\psi''(t)}+\frac{\gamma+N}{\gamma-1}\bigg)\\&=\begin{cases}0, &\,{ if }\,\mathbf{(S_{1})}\,{ holds\, and }\,\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \,{ in }\,\mathbf{(g_{2})},\\0, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})} \,{hold\, with }\,\frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho};\end{cases}\end{split}$$let ξ be a positive constant, thenlimt→0+(−lnψ(t))β(g(ξψ(t))ξNg(ψ(t))−ξ−(γ+N))=−σξ−(γ+N)lnξ,if(S2)and(g3)−(g4)holdwithθ=0in(g3),0,if(S1)holds(or(S2)and(g3)holdwithθ>0);$$\begin{split}&\lim\limits_{t\rightarrow0^{+}}(-\ln \psi(t))^{\beta}\bigg(\frac{g(\xi\psi(t))}{\xi^{N}g(\psi(t))}-\xi^{-(\gamma+N)}\bigg)\\&=\begin{cases}-\sigma\xi^{-(\gamma+N)}\ln\xi, &\,{ if }\,\mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})}\,{-}\mathbf{(g_{4})} \,{ hold\, with }\theta=0 \,{ in }\,\mathbf{(g_{3})},\\0, &\,{ if }\,\mathbf{(S_{1})} \,{ holds\, (or }\,\mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})} \,{ hold \,with }\, \theta>0);\end{cases}\end{split}$$let (S1) hold (or (S2) and (g3) hold with θ(N + 1)>(γ+N)ρ), ξ be a positive constant and ρ ∈ (0, 1], then limt→0+t−ρ(g(ξψ(t))ξNg(ψ(t))−ξ−(γ+N))=0.$ \lim\limits_{t\rightarrow0^{+}}t^{-\rho}\big(\frac{g(\xi \psi(t))}{\xi^{N}g(\psi(t))}-\xi^{-(\gamma+N)}\big)=0. $Proof. (i)-(ii) By the definition of ψ, we obtain that (i)-(ii) hold.(iii)-(iv) It follows by (i) and Lemma 3.3 (iv)-(v) thatlim t → 0 + t ψ ′ ( t ) ψ ( t ) = lim t → 0 + t ( ( N + 1 ) G ( ψ ( t ) ) ) 1 N + 1 ψ ( t ) z := ψ ( t ) _ _ lim z → 0 + φ ( z ) ( ( N + 1 ) G ( z ) ) 1 N + 1 z = N + 1 γ + N $$ \lim\limits_{t\rightarrow0^{+}}\frac{t\psi'(t)}{\psi(t)}=\lim\limits_{t\rightarrow0^{+}}\frac{t((N+1)G(\psi(t)))^{\frac{1}{N+1}}}{\psi(t)}\underline{\underline{z:=\psi(t)}}\lim\limits_{z\rightarrow0^{+}}\frac{\varphi(z)((N+1)G(z))^{\frac{1}{N+1}}}{z}=\frac{N+1}{\gamma+N} $$andlimt→0+ψ′(t)tψ′′(t)=−limz→0+((N+1)G(z))NN+1g(z)φ(z)=−γ+Nγ−1.$$\lim\limits_{t\rightarrow0^{+}}\frac{\psi'(t)}{t\psi''(t)}=-\lim\limits_{z\rightarrow0^{+}}\frac{((N+1)G(z))^{\frac{N}{N+1}}}{g(z)\varphi(z)}=-\frac{\gamma+N}{\gamma-1}.$$(v) We conclude by (iii), Lemma 3.1 (ii) and Lemma 2.5 (iii) that (v) holds.(vi) By using Lemma 3.5, we havelimt→0+t((∫0tL˜(s)sds)NN+1)′(∫0tL˜(s)sds)NN+1=NN+1limt→0+L˜(t)∫0tL˜(s)sds=0.$$\lim\limits_{t\rightarrow0^{+}}\frac{t\big(\big(\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds\big)^{\frac{N}{N+1}}\big)'}{\big(\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds\big)^{\frac{N}{N+1}}}=\frac{N}{N+1}\lim\limits_{t\rightarrow0^{+}}\frac{\tilde{L}(t)}{\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds}=0.$$This implies that(3.17)t↦(∫0tL˜(s)sds)NN+1∈NRVZ0.$$\begin{equation}t\mapsto\bigg(\int\limits_{0}^{t}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\inNRVZ_{0}.\end{equation}$$So, by (3.17), (iii) and Proposition 2.7 we further obtain(3.18)t↦ψ((∫0tL˜(s)sds)NN+1)∈NRVZ0.$$\begin{equation}t\mapsto\psi\bigg(\bigg(\int\limits_{0}^{t}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\inNRVZ_{0}.\end{equation}$$We have by Proposition 2.5 (iii) that (vi) holds.(vii)-(viii) It follows by (i) and Lemma 3.4 (iii) thatlim t → 0 + ( − ln ⁡ ψ ( t ) ) β ( ψ ′ ( t ) t ψ ″ ( t ) + γ + N γ − 1 ) = lim t → 0 + ( − ln ⁡ ψ ( t ) ) β ( − ( ( N + 1 ) G ( ψ ( t ) ) ) N N + 1 g ( ψ ( t ) ) t + γ + N γ − 1 ) z := ψ ( t ) _ _ lim z → 0 + ( − ln ⁡ z ) β ( − ( ( N + 1 ) G ( z ) ) N N + 1 g ( z ) φ ( z ) + γ + N γ − 1 ) = ( N + 1 ) σ ( γ − 1 ) 2 , if ( S 2 ) and ( g 3 ) - ( g 4 ) hold with θ = 0 in ( g 3 ) , 0 , if ( S 1 ) holds (or ( S 2 ) and ( g 3 ) hold with θ > 0 ) , $$ \begin{split} &\lim\limits_{t\rightarrow0^{+}}(-\ln \psi(t))^{\beta}\bigg(\frac{\psi'(t)}{t\psi''(t)}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &=\lim\limits_{t\rightarrow0^{+}}(-\ln \psi(t))^{\beta}\bigg(-\frac{((N+1)G(\psi(t)))^{\frac{N}{N+1}}}{g(\psi(t))t}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &\underline{\underline{z:=\psi(t)}}\lim\limits_{z\rightarrow0^{+}}(-\ln z)^{\beta}\bigg(-\frac{((N+1)G(z))^{\frac{N}{N+1}}}{g(z)\varphi(z)}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &=\begin{cases} \frac{(N+1)\sigma}{(\gamma-1)^{2}}, &\mbox{ if }\mathbf{(S_{2})}\mbox{ and }\mathbf{(g_{3})}\mbox{-}\mathbf{(g_{4})} \mbox{ hold with }\theta=0 \mbox{ in }\mathbf{(g_{3})},\\ 0, &\mbox{ if } \mathbf{(S_{1})} \mbox{ holds (or }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with } \theta>0), \end{cases} \end{split} $$andlim t → 0 + t − ρ ( ψ ′ ( t ) t ψ ″ ( t ) + γ + N γ − 1 ) = lim t → 0 + t − ρ ( − ( ( N + 1 ) G ( ψ ( t ) ) ) N N + 1 g ( ψ ( t ) ) t + γ + N γ − 1 ) z := ψ ( t ) _ _ lim z → 0 + ( φ ( z ) ) − ρ ( − ( ( N + 1 ) G ( z ) ) N N + 1 g ( z ) φ ( z ) + γ + N γ − 1 ) = 0 , if ( S 1 ) holds and γ > ( 1 + ρ ) N + 1 N + 1 − ρ in ( g 2 ) , 0 , if ( S 2 ) and ( g 3 ) hold with θ ( N + 1 ) ρ − N > γ > ( 1 + ρ ) N + 1 N + 1 − ρ . $$ \begin{split} \lim\limits_{t\rightarrow0^{+}}t^{-\rho}\bigg(\frac{\psi'(t)}{t\psi''(t)}+\frac{\gamma+N}{\gamma-1}\bigg) &=\lim\limits_{t\rightarrow0^{+}}t^{-\rho}\bigg(-\frac{((N+1)G(\psi(t)))^{\frac{N}{N+1}}}{g(\psi(t))t}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &\underline{\underline{z:=\psi(t)}}\lim\limits_{z\rightarrow0^{+}}(\varphi(z))^{-\rho}\bigg(-\frac{((N+1)G(z))^{\frac{N}{N+1}}}{g(z)\varphi(z)}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &=\begin{cases} 0, &\mbox{ if }\mathbf{(S_{1})}\mbox{ holds and }\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \mbox{ in }\mathbf{(g_{2})},\\ 0, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with } \frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho}. \end{cases} \end{split} $$(ix)-(x) It follows by Lemma 3.4 (iv) thatlim t → 0 + ( − ln ⁡ ψ ( t ) ) β ( g ( ξ ψ ( t ) ) ξ N g ( ψ ( t ) ) − ξ − ( γ + N ) ) z := ψ ( t ) _ _ lim z → 0 + ( − ln ⁡ z ) β ( g ( ξ z ) ξ N g ( z ) − ξ − ( γ + N ) ) = − σ ξ − ( γ + N ) ln ⁡ ξ , if ( S 2 ) and ( g 3 ) - ( g 4 ) hold with θ = 0 in ( g 3 ) , 0 , if ( S 1 ) holds (or ( S 2 ) and ( g 3 ) hold with θ > 0 ) . $$ \begin{split} &\lim\limits_{t\rightarrow0^{+}}(-\ln \psi(t))^{\beta}\bigg(\frac{g(\xi \psi(t))}{\xi^{N} g(\psi(t))}-\xi^{-(\gamma+N)}\bigg)\\ &\underline{\underline{z:=\psi(t)}}\lim\limits_{z\rightarrow0^{+}}(-\ln z)^{\beta}\bigg(\frac{g(\xi z)}{\xi^{N} g(z)}-\xi^{-(\gamma+N)}\bigg)\\ &=\begin{cases} -\sigma\xi^{-(\gamma+N)}\ln\xi, &\mbox{ if }\mathbf{(S_{2})}\mbox{ and }\mathbf{(g_{3})}\mbox{-}\mathbf{(g_{4})} \mbox{ hold with }\theta=0 \mbox{ in }\mathbf{(g_{3})},\\ 0, &\mbox{ if }\mathbf{(S_{1})} \mbox{ holds (or }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with } \theta>0). \end{cases} \end{split}$$Moreover, if (S1) holds or (S2) and (g3) hold with θ(N + 1)>(γ+N)ρ, then by Lemma 3.4 (iv), we havelimt→0+t−ρg(ξψ(t))ξNg(ψ(t))−ξ−(γ+N)=z:=ψ(t)limz→0+(φ(z))−ρg(ξz)ξNg(z)−ξ−(γ+N)=0.$$\lim\limits_{t \rightarrow 0^{+}} t^{-\rho}\left(\frac{g(\xi \psi(t))}{\xi^{N} g(\psi(t))}-\xi^{-(\gamma+N)}\right) \stackrel{z:=\psi(t)}{=} \lim\limits_{z \rightarrow 0^{+}}(\varphi(z))^{-\rho}\left(\frac{g(\xi z)}{\xi^{N} g(z)}-\xi^{-(\gamma+N)}\right)=0.$$□4The Second Boundary BehaviorIn this section, we prove Theorems 1.1-1.3. We first introduce some lemmas as follows.Lemma 4.1(Lemma 2.1 in [27]) Let Ω be a bounded domain in ℝN with N ≥ 2, and let u,v∈C(Ωˉ)∩C2(Ω)$ u,\,v\in C(\bar{\Omega})\cap C^{2}(\Omega) $. Suppose h(x, t) is defined for x ∈ Ω and t in some interval containingthe ranges of u and v. If the following hold:(i) h is strictly increasing in t for all x ∈ Ω,(ii) the matrix D2v is positive definite in Ω,(iii) det(D2v)=h(x, v) and det (D2u) ⩽ h(x, u), x ∈ Ω,(iv) u=v on ∂Ω,then, we have u≥v in Ω.For any δ>0, letΩδ={x∈Ω:0<d(x)<δ}.$$\Omega_{\delta}=\{x\in\Omega: 0<d(x)<\delta\}.$$Since Ω is Cm-smooth for m≥2, we can always takeδ1 > 0 such that (see Lemmas 14.16 and 14.17 in[18])d∈Cm(Ωδ1)and|∇d(x)|=1,x∈Ωδ1.$$d\in C^{m}(\Omega_{\delta_{1}}) \mbox{ and } |\nablad(x)|=1,\,x\in\Omega_{\delta_{1}}.$$Let xˉ∈∂Ω$ \bar{x}\in\partial\Omega $be the projection of the pointx∈Ωδ1$ x\in\Omega_{\delta_{1}} $to ∂Ω, andκi(xˉ)$ \kappa_{i}(\bar{x}) $(i=1, ···, N − 1) be the principalcurvatures of ∂Ω at xˉ$ \bar{x} $, thenD2(d(x))=diag(−κ1(xˉ)1−d(x)κ1(xˉ),⋅⋅⋅,−κN−1(xˉ)1−d(x)κN−1(xˉ),0).$$D^{2}(d(x))=\mbox{diag}\bigg(\frac{-\kappa_{1}(\bar{x})}{1-d(x)\kappa_{1}(\bar{x})},\cdot\cdot\cdot,\frac{-\kappa_{N-1}(\bar{x})}{1-d(x)\kappa_{N-1}(\bar{x})},0\bigg).$$Lemma 4.2(See the proof of Proposition 2.4 in [27], Proposition 2.1 and Corollary 2.3 in [9]) Let h be a C2-function on (0, δ1), thendet(D2h(d(x)))=(−h′(d(x)))N−1h′′(d(x))∏i=1N−1κi(xˉ)1−d(x)κi(xˉ),x∈Ωδ1.$$\mbox{det}(D^{2}h(d(x)))=(-h'(d(x)))^{N-1}h''(d(x))\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})},\,x\in\Omega_{\delta_{1}}.$$Lemma 4.3(Corollary 2.3 in [20]) Let h be a C2-function on (0, δ1) and Ω be a bounded domain with ∂Ω ∈ Cm for m=2, then, for i=1, ···, N,Si(D2(h(d(x))))=(−h′(d(x)))iSi(ϵ1,⋅⋅⋅,ϵN−1)+(−h′(d(x)))i−1h′′(d(x))Si−1(ϵ1,⋅⋅⋅,ϵN−1),x∈Ωδ1$$S_{i}(D^{2}(h(d(x))))=(-h'(d(x)))^{i}S_{i}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1})+(-h'(d(x)))^{i-1}h''(d(x))S_{i-1}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1}),\,x\in\Omega_{\delta_{1}}$$whereϵj=κj(xˉ)1−κj(xˉ)d(x),j=1,⋅⋅⋅,N−1.$$\epsilon_{j}=\frac{\kappa_{j}(\bar{x})}{1-\kappa_{j}(\bar{x})d(x)},\,j=1,\cdot\cdot\cdot,N-1.$$Definition 4.4(Definition 1.1 in [42]) Let i ∈ {1, ···, N} and Ω be an open bounded subset of ℝN; a function u ∈ C2(Ω) is (strictly) i-convex if Sl(D2u)(>)⩾0 in Ω for l=1, ···, i. In particular, if i=N, then we say that u is (strictly) convex in Ω.4.1Proof of Theorem 1.1Next, we prove Theorem 1.1 and we first show some preliminaries as follows.Fix ε>0 and letw±(d(x))=ξ±ψ(K(d(x)))(1+(C±±ε)(−lnd(x))−β),x∈Ωδ1,$$w_{\pm}(d(x))=\xi_{\pm}\psi(K(d(x)))\big(1+(C_{\pm}\pm\varepsilon)(-\lnd(x))^{-\beta}\big),\,x\in\Omega_{\delta_{1}},$$where ξ ± and C ± are in Theorem 1.1. By the Lagrange’s mean value theorem, we obtain that there exist λ ± ∈ (0, 1) and(4.1)Θ±(d(x))=ξ±ψ(K(d(x)))(1+λ±(C±±ε)(−lnd(x))−β)$$\begin{equation}\Theta_{\pm}(d(x))=\xi_{\pm}\psi(K(d(x)))\big(1+\lambda_{\pm}(C_{\pm}\pm\varepsilon)(-\lnd(x))^{-\beta}\big)\end{equation}$$such that for any x∈Ωδ1$ x\in\Omega_{\delta_{1}} $g(w±(d(x)))=g(ξ±ψ(K(d(x))))+ξ±ψ(K(d(x)))g′(Θ±(d(x)))(C±±ε)(−lnd(x))−β.$$\begin{split}g(w_{\pm}(d(x)))&=g(\xi_{\pm}\psi(K(d(x))))+\xi_{\pm}\psi(K(d(x)))g'(\Theta_{\pm}(d(x)))(C_{\pm}\pm\varepsilon)(-\lnd(x) )^{-\beta}.\end{split}$$Since g ∈ NRVZ−γ, we have by Proposition 2.2 that(4.2)limd(x)→0g(ξ±ψ(K(d(x))))g(Θ±(d(x)))=limd(x)→0g′(ξ±ψ(K(d(x))))g′(Θ±(d(x)))=1.$$\begin{equation}\lim\limits_{d(x)\rightarrow0}\frac{g(\xi_{\pm}\psi(K(d(x))))}{g(\Theta_{\pm}(d(x)))}=\lim\limits_{d(x)\rightarrow0}\frac{g'(\xi_{\pm}\psi\big(K(d(x))))}{g'(\Theta_{\pm}(d(x)))}=1.\end{equation}$$Moreover, by the definitions of ξ ± and C ± , we can take a sufficiently small positive constant still denoted by δ1 such that(4.3)w+(d(x))≥w−(d(x)),x∈Ωδ1.$$\begin{equation}w_{+}(d(x))\geq w_{-}(d(x)),\,x\in\Omega_{\delta_{1}}.\end{equation}$$Proof. Our proof is divided into two steps and the outline of the proof is given as below.• In Step 1, for fixed ε>0 andx∈Ωδ1$ x\in\Omega_{\delta_{1}} $, we first give some functions I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) (which are corresponding to ε>0), and then by detailed calculation we will show that there exists a sufficiently small positive constant δε<δ1 such that(4.4)I1+(d(x))+I2+(d(x))+I3+(d(x))>0,x∈Ωδε$$\begin{equation}I_{1+}(d(x))+I_{2+}(d(x))+I_{3+}(d(x))>0,\,x\in\Omega_{\delta_{\varepsilon}}\end{equation}$$and(4.5)I1−(d(x))+I2−(d(x))+I3−(d(x))<0,x∈Ωδε.$$\begin{equation}I_{1-}(d(x))+I_{2-}(d(x))+I_{3-}(d(x))<0,\,x\in\Omega_{\delta_{\varepsilon}}.\end{equation}$$• In Step 2, we will define two functions u_ε,u‾ε$ \underline{u}_{\varepsilon}, \overline{u}_{\varepsilon} $inΩδε$ \Omega_{\delta_{\varepsilon}} $and show they are sub-and super-solutions of Eq. (1.1) in Ωδε$ \Omega_{\delta_{\varepsilon}} $by (4.4) and (4.5), respectively. In particular, we will show u_ε$ \underline{u}_{\varepsilon} $is strictly convex inΩδε$ \Omega_{\delta_{\varepsilon}} $. Finally, we will establish the second boundary behavior of the unique strictly convex solution to problem (1.1) by using Lemma 4.1.By the above analysis, we see that how to get (4.4)-(4.5) is the key of the research.Step 1.We first define functions I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as follows.For fixed ε>0 and ∀x∈Ωδ1$ \forall\,x\in\Omega_{\delta_{1}} $, we defineI1±(d(x))=(−lnd(x))β[(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ±−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))];I2±(d(x))=C±[N(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)A±∏i=1N−1(1−d(x)κi(xˉ))−1−ξ±ψ(K(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))g′(Θ±(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))]±ε[N(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ+∏i=1N−1(1−d(x)κi(xˉ))−1−ξ±ψ(K(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))g′(Θ±(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))];I3±(d(x))=(−lnd(x))β(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)×mˆ±∑i=1N−1CN−1i(−1)i+1(m±d(x))i(1−d(x)m±)N−1+ζ±(d(x))(−lnd(x))−β∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)−(B0±ε)g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))(d(x))μ(−lnd(x))β−(B0±ε)(C±±ε)ξ±ψ(K(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))g′(Θ±(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))(d(x))μ,$$\begin{split}&I_{1\pm}(d(x))=(-\lnd(x))^{\beta}\bigg[\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\hat{m}_{\pm}\\&-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg];\\&I_{2\pm}(d(x))=C_{\pm}\bigg[N\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\mathscr{A}_{\pm}\prod\limits_{i=1}^{N-1}(1-d(x)\kappa_{i}(\bar{x}))^{-1}\\&-\frac{\xi_{\pm}\psi(K(d(x)))g'(\xi_{\pm}\psi(K(d(x))))}{g(\xi_{\pm}\psi(K(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\xi_{\pm}\psi(K(d(x))))}\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg]\\&\pm\varepsilon\bigg[N\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\hat{m}_{+}\prod\limits_{i=1}^{N-1}(1-d(x)\kappa_{i}(\bar{x}))^{-1}\\&-\frac{\xi_{\pm}\psi(K(d(x)))g'(\xi_{\pm}\psi(K(d(x))))}{g(\xi_{\pm}\psi(K(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\xi_{\pm}\psi(K(d(x))))}\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg];\\&I_{3\pm}(d(x))=(-\lnd(x))^{\beta}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\\&\times\frac{\hat{m}_{\pm}\sum_{i=1}^{N-1}C_{N-1}^{i}(-1)^{i+1}(m_{\pm}d(x))^{i}}{(1-d(x)m_{\pm})^{N-1}}+\frac{\zeta_{\pm}(d(x))}{(-\lnd(x))^{-\beta}}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\\&-(B_{0}\pm\varepsilon)\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}(d(x))^{\mu}(-\lnd(x))^{\beta}\\&-(B_{0}\pm\varepsilon)(C_{\pm}\pm\varepsilon)\frac{\xi_{\pm}\psi(K(d(x)))g'(\xi_{\pm}\psi(K(d(x))))}{g(\xi_{\pm}\psi(K(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\xi_{\pm}\psi(K(d(x))))}\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}(d(x))^{\mu},\end{split}$$where(S2) holds and θ=0k∈Λ1A±=mˆ±, if A+≤0 and A−≤0,A±=mˆ∓, if A+≥0 and A−≥0,A+=A−=mˆ−, if A+>0 and A−<0,A+=A−=mˆ+, if A+<0 and A−>0;k∈Λ2,βA±=mˆ±, if A++B≤0 and A++B≤0,A±=mˆ∓, if A++B≥0 and A++B≥0,A+=A−=mˆ−, if A++B>0 and A++B<0,A+=A−=mˆ+, if A++B<0 and A++B>0;(S1) holdsk∈Λ1,A±=mˆ±;k∈Λ2,β,A+=mˆ−, if B≥0,A−=mˆ+, if B<0;(S2) holds and θ>0k∈Λ2,β,A+=mˆ−, if B≥0,A−=mˆ+, if B<0,$$\begin{cases}\mathbf{(S_{2})}\ \mbox{ holds and }\theta=0&\begin{cases}k\in\Lambda_{1}&\begin{cases}\mathscr{A}_{\pm}=\hat{m}_{\pm},\, &\mbox{ if }A_{+}\leq0 \mbox{ and }A_{-}\leq0,\\\mathscr{A}_{\pm}=\hat{m}_{\mp},\,&\mbox{if }A_{+}\geq0 \mbox{ and }A_{-}\geq0,\\\mathscr{A}_{+}=\mathscr{A}_{-}=\hat{m}_{-},\,&\mbox{if }A_{+}>0 \mbox{ and }A_{-}<0,\\\mathscr{A}_{+}=\mathscr{A}_{-}=\hat{m}_{+},\,&\mbox{if }A_{+}<0 \mbox{ and }A_{-}>0;\\\end{cases}\\k\in\Lambda_{2,\beta}&\begin{cases}\mathscr{A}_{\pm}=\hat{m}_{\pm},\, &\mbox{ if }\mathfrak{A}_{+}+\mathfrak{B}\leq0 \mbox{ and }\mathfrak{A}_{+}+\mathfrak{B}\leq0,\\\mathscr{A}_{\pm}=\hat{m}_{\mp},\,&\mbox{if }\mathfrak{A}_{+}+\mathfrak{B}\geq0 \mbox{ and }\mathfrak{A}_{+}+\mathfrak{B}\geq0,\\\mathscr{A}_{+}=\mathscr{A}_{-}=\hat{m}_{-},\,&\mbox{if }\mathfrak{A}_{+}+\mathfrak{B}>0 \mbox{ and }\mathfrak{A}_{+}+\mathfrak{B}<0,\\\mathscr{A}_{+}=\mathscr{A}_{-}=\hat{m}_{+},\,&\mbox{if }\mathfrak{A}_{+}+\mathfrak{B}<0 \mbox{ and }\mathfrak{A}_{+}+\mathfrak{B}>0;\\\end{cases}\end{cases}\\\mathbf{(S_{1})} \mbox{ holds}&\begin{cases}k\in\Lambda_{1}, &\mathscr{A}_{\pm}=\hat{m}_{\pm};\\k\in\Lambda_{2,\beta}, &\begin{cases}\mathscr{A}_{+}=\hat{m}_{-},\,&\mbox{ if }\mathfrak{B}\geq0,\\\mathscr{A}_{-}=\hat{m}_{+},\,&\mbox{ if }\mathfrak{B}<0;\end{cases}\end{cases}\\\mathbf{(S_{2})}\ \mbox{ holds and }\theta>0&\begin{cases}k\in\Lambda_{2,\beta}, &\begin{cases}\mathscr{A}_{+}=\hat{m}_{-},\,&\mbox{ if }\mathfrak{B}\geq0,\\\mathscr{A}_{-}=\hat{m}_{+},\,&\mbox{ if }\mathfrak{B}<0,\end{cases}\end{cases}\end{cases}$$(4.6)m+:=min{minxˉ∈∂Ωκi(xˉ):i=1,⋅⋅⋅,N−1} and m−:=max{maxxˉ∈∂Ωκi(xˉ):i=1,⋅⋅⋅,N−1}$$\begin{equation}m_{+}:=\min\{\min\limits_{\bar{x}\in\partial\Omega}\kappa_{i}(\bar{x}):i=1,\cdot\cdot\cdot,N-1\}\mbox{ and }m_{-}:=\max\{\max\limits_{\bar{x}\in\partial\Omega}\kappa_{i}(\bar{x}):i=1,\cdot\cdot\cdot,N-1\}\end{equation}$$andζ±(d(x))=(C±±ε)2(N−1)(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(−lnd(x))−2β+2β(C±±ε)ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)(−lnd(x))−β−1×(1+(C±±ε)(N−1)(−lnd(x))−β+ν±(d(x)))−β(C±±ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2(−lnd(x))−β−1×(1−(β+1)(−lnd(x))−1)(1+(C±±ε)(N−1)(−lnd(x))−β+ν±(d(x)))+ν±(d(x))(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(1+(C±±ε)(−lnd(x))−β),$$\begin{align*}\zeta_{\pm}(d(x))&=(C_{\pm}\pm\varepsilon)^{2}(N-1)\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)(-\lnd(x))^{-2\beta}\\&+2\beta(C_{\pm}\pm\varepsilon)\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{k(d(x))d(x)}(-\lnd(x))^{-\beta-1}\\&\times\big(1+(C_{\pm}\pm\varepsilon)(N-1)(-\lnd(x))^{-\beta}+\nu_{\pm}(d(x))\big)\\&-\beta(C_{\pm}\pm\varepsilon)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}(-\lnd(x))^{-\beta-1}\\&\times\big(1-(\beta+1)(-\lnd(x))^{-1}\big)\big(1+(C_{\pm}\pm\varepsilon)(N-1)(-\lnd(x))^{-\beta}+\nu_{\pm}(d(x))\big)\\&+\nu_{\pm}(d(x))\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\big(1+(C_{\pm}\pm\varepsilon)(-\lnd(x))^{-\beta}\big),\end{align*}$$where A ± are given by (1.8), A±$ \mathfrak{A}_{\pm} $and B$ \mathfrak{B} $are given by (1.11) andν±(d(x))=R2±(d(x))+∑i=2N−1CN−1i(R1±(d(x))+R2±(d(x)))i,$$\nu_{\pm}(d(x))=\mathfrak{R}_{2\pm}(d(x))+\sum_{i=2}^{N-1}C_{N-1}^{i}\big(\mathfrak{R}_{1\pm}(d(x))+\mathfrak{R}_{2\pm}(d(x))\big)^{i},$$where(4.7)CN−1i=(N−1)!i!(N−1−i)!,R1±(d(x))=(C±±ε)(−lnd(x))−β,$$\begin{equation}C_{N-1}^{i}=\frac{(N-1)!}{i!(N-1-i)!},\,\,\,\mathfrak{R}_{1\pm}(d(x))=(C_{\pm}\pm\varepsilon)(-\lnd(x))^{-\beta},\end{equation}$$andR2±(d(x))=(N−1)β(C±±ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)(−lnd(x))−β−1.$$\mathfrak{R}_{2\pm}(d(x))=(N-1)\beta(C_{\pm}\pm\varepsilon)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{k(d(x))d(x)}(-\lnd(x))^{-\beta-1}.$$Next, we prove(4.8)limd(x)→0(I1±(d(x))+I2±(d(x))+I3±(d(x)))=±ε(γ+Nmˆ+mˆ±)ξ±−(γ+N).$$\begin{equation}\lim_{d(x)\rightarrow0}(I_{1\pm}(d(x))+I_{2\pm}(d(x))+I_{3\pm}(d(x)))=\pm\varepsilon\bigg(\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{\pm}}\bigg)\xi_{\pm}^{-(\gamma+N)}.\end{equation}$$To prove (4.8), we calculate the limits of I1 ± (d(x)),I2 ± (d(x)) and I3 ± (d(x)) as d(x) → 0.• First, by Lemma 3.1 (ii), Lemma 3.6 (v), (vii) and (ix), we obtain that(4.9)limd(x)→0I1±(d(x))=ϖ1±+ϖ2±, if (S2) and (g3) hold with θ=0,k∈Λ1,ϖ1±+ϖ2±+(γ+N)E2,kmˆ±γ−1, if (S2) and (g3) hold with θ=0,k∈Λ2,β,γ+Nγ−1E2,kmˆ±, if (S2) and (g3) hold with θ>0,k∈Λ2,β,γ+Nγ−1E2,kmˆ±, if (S1) holds,k∈Λ2,β,0, if (S1) holds,k∈Λ1,$$\begin{equation}\lim\limits_{d(x)\rightarrow0}I_{1\pm}(d(x))=\begin{cases}\varpi_{1\pm}+\varpi_{2\pm}, &\mbox{ if } \mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with }\theta=0, k\in\Lambda_{1}, \\\varpi_{1\pm}+\varpi_{2\pm}+\frac{(\gamma+N)E_{2,k}\hat{m}_{\pm}}{\gamma-1}, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with }\theta=0,\, k\in\Lambda_{2, \beta},\\\frac{\gamma+N}{\gamma-1}E_{2,k}\hat{m}_{\pm}, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with}\theta>0,k\in\Lambda_{2, \beta},\\\frac{\gamma+N}{\gamma-1}E_{2,k}\hat{m}_{\pm},&\mbox{ if }\mathbf{(S_{1})}\mbox{ holds}, k\in\Lambda_{2, \beta},\\0, &\mbox{ if }\mathbf{(S_{1})} \mbox{ holds}, k\in\Lambda_{1},\\\end{cases}\end{equation}$$whereϖ1±=((γ+N)DkN+1)β(N+1)(1−Dk)σmˆ±(γ−1)2andϖ2±=((γ+N)DkN+1)βσlnξ±ξ±γ+N.$$\varpi_{1\pm}=\bigg(\frac{(\gamma+N)D_{k}}{N+1}\bigg)^{\beta}\frac{(N+1)(1-D_{k})\sigma\hat{m}_{\pm}}{(\gamma-1)^{2}} \mbox{ and }\varpi_{2\pm}=\bigg(\frac{(\gamma+N)D_{k}}{N+1}\bigg)^{\beta}\frac{\sigma\ln\xi_{\pm}}{\xi_{\pm}^{\gamma+N}}.$$• Second, by (4.2), Lemma 3.1 (ii), Lemma 3.6 (iv) and thechoices of ξ ± , C ± in Theorem 1.1, we obtain(4.10)limd(x)→0I2±(d(x))=C±[(γ+N)Dk−(N+1)]Nγ−1A±+γξ±−(γ+N)±ε(γ+Nmˆ+mˆ±)ξ±−(γ+N).$$\begin{equation}\lim_{d(x)\rightarrow0}I_{2\pm}(d(x))=C_{\pm}\frac{[(\gamma+N)D_{k}-(N+1)]N}{\gamma-1}\mathscr{A}_{\pm}+\gamma\xi_{\pm}^{-(\gamma+N)}\pm\varepsilon\bigg(\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{\pm}}\bigg)\xi_{\pm}^{-(\gamma+N)}.\end{equation}$$• Third, by Lemma 3.1 (ii), Lemma 3.6 (iii)-(iv) and a straightforward calculation, we obtain(4.11)limd(x)→0I3±(d(x))=0.$$\begin{equation}\lim_{d(x)\rightarrow0}I_{3\pm}(d(x))=0.\end{equation}$$Combining (4.9) and (4.10)-(4.11), we obtain (4.8) holds. By (b1)-(b2) and (4.8), we see that there exists asufficiently small constant δε<δ1$ \delta_{\varepsilon}<\delta_{1} $such that(4.12)kN+1(d(x))1+B0−ε(d(x))μ≤b(x)≤kN+1(d(x))1+B0+ε(d(x))μ,x∈Ωδε$$\begin{equation}k^{N+1}(d(x))\left(1+\left(B_{0}-\varepsilon\right)(d(x))^{\mu}\right) \leq b(x) \leq k^{N+1}(d(x))\left(1+\left(B_{0}+\varepsilon\right)(d(x))^{\mu}\right), x \in \Omega_{\delta_{\varepsilon}}\end{equation}$$and (4.4)-(4.5) hold.Step 2. Letu_ε(x)=−ξ+ψ(K(d(x)))(1+(C++ε)(−lnd(x))−β),x∈Ωδε.$$\underline{u}_{\varepsilon}(x)=-\xi_{+}\psi(K(d(x)))\big(1+(C_{+}+\varepsilon)(-\lnd(x))^{-\beta}\big),\,x\in\Omega_{\delta_{\varepsilon}}.$$Theng(−u_ε(x))=g(ξ+ψ(K(d(x))))+ξ+ψ(K(d(x)))g′(Θ+(d(x)))(C++ε)(−lnd(x))−β,x∈Ωδε,$$\begin{split}g(-\underline{u}_{\varepsilon}(x))&=g(\xi_{+}\psi(K(d(x))))+\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(C_{+}+\varepsilon)(-\lnd(x) )^{-\beta},\,x\in\Omega_{\delta_{\varepsilon}},\end{split}$$where Θ+(d(x))$\Theta_{+}(d(x))$is given by (4.1). By Lemma 4.2, we have for any x∈Ωδε$x \in \Omega_{\delta_{\varepsilon}}$(4.13)det D 2 u _ ε ( x ) − b ( x ) g − u _ ε ( x ) ≥ − ξ + N ψ ′ ( K ( d ( x ) ) ) N − 1 ψ ′ ′ ( K ( d ( x ) ) ) k N + 1 ( d ( x ) ) × 1 + C + + ε ( N − 1 ) ( − ln ⁡ d ( x ) ) − β + v + ( d ( x ) ) × ψ ′ ( K ( x ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 1 + C + + ε ( − ln ⁡ d ( x ) ) − β + 2 β C + + ε × ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ( d ( x ) ) d ( x ) ( − ln ⁡ d ( x ) ) − β − 1 − β C + + ε ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 ( − ln ⁡ d ( x ) ) − β − 1 × 1 − ( β + 1 ) ( − ln ⁡ d ( x ) ) − 1 ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − k N + 1 ( d ( x ) ) 1 + B 0 + ε ( d ( x ) ) μ × g ξ + ψ ( K ( d ( x ) ) ) + ξ + ψ ( K ( d ( x ) ) ) g ′ Θ + ( d ( x ) ) C + + ε ( − ln ⁡ d ( x ) ) − β ≥ − ξ + N ψ ′ ( K ( d ( x ) ) ) N − 1 ψ ′ ′ ( K ( d ( x ) ) ) k N + 1 ( d ( x ) ) ( − ln ⁡ d ( x ) ) − β ( − ln ⁡ d ( x ) ) β l × ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 ∏ i = 1 N − 1 m ^ + 1 − d ( x ) m + N − 1 − m ^ + + m ^ + − g ( ξ + ψ ( K ( d ( x ) ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) + C + + ε N ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − ξ + ψ ( K ( d ( x ) ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ξ + ψ ( K ( d ( x ) ) ) × g ′ Θ + ( d ( x ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ξ + ψ ( K ( d ( x ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) + ζ + ( d ( x ) ) ( − ln ⁡ d ( x ) ) − β ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − B 0 + ε g ξ + ψ ( K ( d ( x ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) ( d ( x ) ) μ ( − ln ⁡ d ( x ) ) β − B 0 + ε C + + ε × ξ + ψ ( K ( d ( x ) ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ( ξ + ψ ( K ( d ( x ) ) ) ) g ′ Θ + ( d ( x ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ξ + ψ ( K ( d ( x ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) ( d ( x ) ) μ ≥ − ξ + N ψ ′ ( K ( d ( x ) ) ) N − 1 ψ ′ ′ ( K ( d ( x ) ) ) k N + 1 ( d ( x ) ) ( − ln ⁡ d ( x ) ) − β ∑ i = 1 3 I i + ( d ( x ) ) > 0 , $$\begin{equation}\begin{array}{l}\operatorname{det}\left(D^{2} \underline{u}_{\varepsilon}(x)\right)-b(x) g\left(-\underline{u}_{\varepsilon}(x)\right) \geq-\xi_{+}^{N}\left(\psi^{\prime}(K(d(x)))\right)^{N-1} \psi^{\prime \prime}(K(d(x))) k^{N+1}(d(x)) \\ \times\left(1+\left(C_{+}+\varepsilon\right)(N-1)(-\ln d(x))^{-\beta}+v_{+}(d(x))\right) \\ \times\left[\left(\frac{\left.\psi^{\prime}(K(x))\right)}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\left(1+\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}\right)+2 \beta\left(C_{+}+\varepsilon\right)\right. \\ \times \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{k(d(x)) d(x)}(-\ln d(x))^{-\beta-1} \\ -\beta\left(C_{+}+\varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\left.\psi^{\prime}(K(d(x)))\right)}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}(-\ln d(x))^{-\beta-1} \\ \left.\times\left(1-(\beta+1)(-\ln d(x))^{-1}\right)\right] \prod\limits_{i=1}^{N-1} \frac{\kappa_{i}(\bar{x})}{1-d(x) \kappa_{i}(\bar{x})}-k^{N+1}(d(x))\left(1+\left(B_{0}+\varepsilon\right)(d(x))^{\mu}\right) \\ \times\left[g\left(\xi_{+} \psi(K(d(x)))\right)+\xi+\psi(K(d(x))) g^{\prime}\left(\Theta_{+}(d(x))\right)\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}\right] \\ \geq-\xi_{+}^{N}\left(\psi^{\prime}(K(d(x)))\right)^{N-1} \psi^{\prime \prime}(K(d(x))) k^{N+1}(d(x))(-\ln d(x))^{-\beta}\left\{(-\ln d(x))^{\beta}\right. \\ {l}\times\left[\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\left(\prod\limits_{i=1}^{N-1} \frac{\hat{m}_{+}}{\left(1-d(x) m_{+}\right)^{N-1}}-\hat{m}_{+}+\hat{m}_{+}\right)\right. \\ \left.-\frac{g(\xi+\psi(K(d(x))))}{\xi_{+}^{N} g(\psi(K(d(x))))}\right]+\left(C_{+}+\varepsilon\right)\left[N\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\right. \\ \times \prod\limits_{i=1}^{N-1} \frac{\kappa_{i}(\bar{x})}{1-d(x) \kappa_{i}(\bar{x})}-\frac{\xi_{+} \psi(K(d(x))) g^{\prime}\left(\xi_{+} \psi(K(d(x)))\right)}{g\left(\xi_{+} \psi(K(d(x)))\right)} \\ \left.\times \frac{g^{\prime}\left(\Theta_{+}(d(x))\right)}{g^{\prime}\left(\xi_{+} \psi(K(d(x)))\right)} \frac{g\left(\xi_{+} \psi(K(d(x)))\right)}{\xi_{+}^{N} g(\psi(K(d(x))))}\right]+\frac{\zeta_{+}(d(x))}{(-\ln d(x))^{-\beta}} \prod\limits_{i=1}^{N-1} \frac{\kappa_{i}(\bar{x})}{1-d(x) \kappa_{i}(\bar{x})} \\ -\left(B_{0}+\varepsilon\right) \frac{g\left(\xi_{+} \psi(K(d(x)))\right)}{\xi_{+}^{N} g(\psi(K(d(x))))}(d(x))^{\mu}(-\ln d(x))^{\beta}-\left(B_{0}+\varepsilon\right)\left(C_{+}+\varepsilon\right) \\ \left.\times \frac{\xi_{+} \psi(K(d(x))) g^{\prime}\left(\xi_{+} \psi(K(d(x)))\right)}{g(\xi+\psi(K(d(x))))} \frac{g^{\prime}\left(\Theta_{+}(d(x))\right)}{g^{\prime}\left(\xi_{+} \psi(K(d(x)))\right)} \frac{g\left(\xi_{+} \psi(K(d(x)))\right)}{\xi_{+}^{N} g(\psi(K(d(x))))}(d(x))^{\mu}\right\} \\ \geq-\xi_{+}^{N}\left(\psi^{\prime}(K(d(x)))\right)^{N-1} \psi^{\prime \prime}(K(d(x))) k^{N+1}(d(x))(-\ln d(x))^{-\beta} \sum_{i=1}^{3} I_{i+}(d(x))>0,\end{array} \end{equation}$$i.e., u_ε$\underline{u}_{\varepsilon}$is a subsolution of Eq. (1.1) in Ωδε$\Omega_{\delta_{\varepsilon}}$. Moreover, it follows by Lemma 4.3 that for i = 1, ..., NSiD2u_ε(x)=ξ+iψ′(K(d(x)))i1+C++ε(−lnd(x))−β+βC++εψ(K(d(x)))ψ′(K(d(x)))K(d(x))×K(d(x))k(d(x))d(x)(−lnd(x))−β−1iki(d(x))Siϵ1,⋯,ϵN−1−ξ+iψ′(K(d(x)))i−1ψ′′(K(d(x)))×ki+1(d(x))1+C++ε(−lnd(x))−β+βC++εψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)×(−lnd(x))−β−1i−1ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1×1+C++ε(−lnd(x))−β+2βC++εψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)(−lnd(x))−β−1−βC++εψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))2(−lnd(x))−β−1×1−(β+1)(−lnd(x))−1Si−1ϵ1,⋯,ϵN−1$$\begin{equation}\begin{array}{l}&S_{i}\left(D^{2} \underline{u}_{\varepsilon}(x)\right)=\xi_{+}^{i}\left(\psi^{\prime}(K(d(x)))\right)^{i}\left(1+\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}+\beta\left(C_{+}+\varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))}\right. \\ & \left.\times \frac{K(d(x))}{k(d(x)) d(x)}(-\ln d(x))^{-\beta-1}\right)^{i} k^{i}(d(x)) S_{i}\left(\epsilon_{1}, \cdots, \epsilon_{N-1}\right)-\xi_{+}^{i}\left(\psi^{\prime}(K(d(x)))\right)^{i-1} \psi^{\prime \prime}(K(d(x))) \\ & \times k^{i+1}(d(x))\left(1+\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}+\beta\left(C_{+}+\varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{k(d(x)) d(x)}\right. \\ & \left.\times(-\ln d(x))^{-\beta-1}\right)^{i-1}\left[\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\right. \\ & \times\left(1+\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}\right)+2 \beta\left(C_{+}+\varepsilon\right) \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{k(d(x)) d(x)}(-\ln d(x))^{-\beta-1} \\ & -\beta\left(C_{+}+\varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}(-\ln d(x))^{-\beta-1} \\ & \left.\times\left(1-(\beta+1)(-\ln d(x))^{-1}\right)\right] S_{i-1}\left(\epsilon_{1}, \cdots, \epsilon_{N-1}\right)\end{array} \end{equation}$$This implies that we can adjust the above positive constant δε$\delta_{\varepsilon}$such that for any x∈Ωδε$x \in \Omega_{\delta_{\varepsilon}}$S i ( D 2 u _ ε ( x ) ) > 0 for i = 1 , ⋅ ⋅ ⋅ , N . $$S_{i}(D^{2}\underline{u}_{\varepsilon}(x))>0 \,\mbox{for}\,i=1,\cdot\cdot\cdot,N.$$We obtain by Definition 4.4 that D2u_ε$D^{2} \underline{u}_{\varepsilon}$is positive definite in Ωδε.$\Omega_{\delta_{\varepsilon}}.$Letu‾ε(x)=−ξ−ψ(K(d(x)))(1+(C−−ε)(−lnd(x))−β),x∈Ωδε.$$\overline{u}_{\varepsilon}(x)=-\xi_{-}\psi(K(d(x)))\big(1+(C_{-}-\varepsilon)(-\lnd(x))^{-\beta}\big),\,x\in\Omega_{\delta_{\varepsilon}}.$$By the same calculation as (4.13), we obtaindetD2uˉε(x)−b(x)g−uˉε(x)≤−ξ−Nψ′(K(d(x)))N−1ψ′′(K(d(x)))kN+1(d(x))(−lnd(x))−β∑i=13Ii−(d(x))<0,$$\begin{equation}\begin{array}{l}\operatorname{det}\left(D^{2} \bar{u}_{\varepsilon}(x)\right)-b(x) g\left(-\bar{u}_{\varepsilon}(x)\right) \\ \leq-\xi_{-}^{N}\left(\psi^{\prime}(K(d(x)))\right)^{N-1} \psi^{\prime \prime}(K(d(x))) k^{N+1}(d(x))(-\ln d(x))^{-\beta} \sum_{i=1}^{3} I_{i-}(d(x))<0,\end{array} \end{equation}$$i.e. uˉε$\bar{u}_{\varepsilon}$is a supersolution of Eq. (1.1) inΩδε.$\Omega_{\delta_{\varepsilon}}.$Let u be the unique strictly convex solution to problem (1.1). Then, there exists a large constant M > 0 such thatu_ε(x)−Md(x)≤u(x)≤u‾ε(x)+Md(x),x∈{x∈Ω:d(x)=δε}.$$\underline{u}_{\varepsilon}(x)-Md(x)\leq u(x)\leq\overline{u}_{\varepsilon}(x)+Md(x),\,x\in\{x\in\Omega:d(x)=\delta_{\varepsilon}\}.$$We assert that(4.14)u(x)≥u_ε(x)−Md(x),x∈Ωδε$$\begin{equation}u(x)\geq\underline{u}_{\varepsilon}(x)-Md(x),\,x\in\Omega_{\delta_{\varepsilon}}\end{equation}$$and(4.15)u(x)≤u‾ε(x)+Md(x),x∈Ωδε.$$\begin{equation}u(x)\leq\overline{u}_{\varepsilon}(x)+Md(x),\,x\in\Omega_{\delta_{\varepsilon}}.\end{equation}$$Since D2u_ε$ D^{2}\underline{u}_{\varepsilon} $is positive definite in Ωδε$ \Omega_{\delta_{\varepsilon}} $and D2(−Md (x)) is positive semidefinite in Ωδε$ \Omega_{\delta_{\varepsilon}} $, we have by the Minkowski inequality that D2(u_ε(x)−Md(x))$ D^{2}(\underline{u}_{\varepsilon}(x)-Md(x)) $is positive definite in Ωδε$ \Omega_{\delta_{\varepsilon}} $anddet(D2(u_ε(x)−Md(x)))≥det(D2u_ε(x))≥b(x)g(−u_ε(x))≥b(x)g(−u_ε(x)+Md(x)),x∈Ωδε.$$\mbox{det}(D^{2}(\underline{u}_{\varepsilon}(x)-Md(x)))\geq\mbox{det}(D^{2}\underline{u}_{\varepsilon}(x))\geqb(x)g(-\underline{u}_{\varepsilon}(x))\geqb(x)g(-\underline{u}_{\varepsilon}(x)+Md(x)),\,x\in\Omega_{\delta_{\varepsilon}}.$$Similarly, we have D2(u(x)−Md (x)) is positive definite in Ωδε$ \Omega_{\delta_{\varepsilon}} $anddet(D2u(x)−Md(x))≥det(D2u(x))=b(x)g(−u(x))≥b(x)g(−u(x)+Md(x)).$$\mbox{det}(D^{2}u(x)-Md(x))\geq\mbox{det}(D^{2}u(x))=b(x)g(-u(x))\geq b(x)g(-u(x)+Md(x)).$$By Lemma 4.1, we see that (4.14)-(4.15) hold. Hence, for any x∈Ωδε$ x\in\Omega_{\delta_{\varepsilon}} $C++ε+Md(x)(−lnd(x))βξ+ψ(K(d(x)))≥−u(x)ξ+ψ(K(d(x)))−1(−lnd(x))β;C−−ε−Md(x)(−lnd(x))βξ−ψ(K(d(x)))≤−u(x)ξ−ψ(K(d(x)))−1(−lnd(x))β.$$\begin{equation}\begin{array}{l}C_{+}+\varepsilon+\frac{M d(x)(-\ln d(x))^{\beta}}{\xi+\psi(K(d(x)))} \geq\left(\frac{-u(x)}{\xi+\psi(K(d(x)))}-1\right)(-\ln d(x))^{\beta}; \\ C_{-}-\varepsilon-\frac{M d(x)(-\ln d(x))^{\beta}}{\xi_{-} \psi(K(d(x)))} \leq\left(\frac{-u(x)}{\xi-\psi(K(d(x)))}-1\right)(-\ln d(x))^{\beta}.\end{array} \end{equation}$$Since (γ+N)Dk−(N + 1)>0, we conclude from Lemma 3.1 (ii), Lemma 3.6 (iii), Proposition 2.7 and Proposition 2.5 (ii) thatC++ε≥lim supd(x)→0−u(x)ξ+ψ(K(d(x)))−1(−lnd(x))β;C−−ε≤lim infd(x)→0−u(x)ξ−ψ(K(d(x)))−1(−lnd(x))β.$$\begin{equation}\begin{array}{l}C_{+}+\varepsilon \geq \limsup _{d(x) \rightarrow 0}\left(\frac{-u(x)}{\xi+\psi(K(d(x)))}-1\right)(-\ln d(x))^{\beta}; \\ C_{-}-\varepsilon \leq \liminf _{d(x) \rightarrow 0}\left(\frac{-u(x)}{\xi_{-} \psi(K(d(x)))}-1\right)(-\ln d(x))^{\beta}.\end{array} \end{equation}$$Letting ε → 0, the proof is finished. □4.2Proof of Theorem 1.2Now, we prove Theorem 1.2. For fixed ε>0, we definew±(d(x))=ξ±ψ(K(d(x)))(1+(C˜±±ε)(d(x))μ),x∈Ωδ1,$$w_{\pm}(d(x))=\xi_{\pm}\psi(K(d(x)))\big(1+(\tilde{C}_{\pm}\pm\varepsilon)(d(x))^{\mu}\big),\,x\in\Omega_{\delta_{1}},$$where ξ ± and C˜±$ \tilde{C}_{\pm} $are given in Theorem 1.2. By the Lagrange’s mean value theorem, we see that there exist λ ± ∈ (0, 1) and(4.16)Θ±(d(x))=ξ±ψ(K(d(x)))(1+λ±(C˜±±ε)(d(x))μ)$$\begin{equation}\Theta_{\pm}(d(x))=\xi_{\pm}\psi(K(d(x)))\big(1+\lambda_{\pm}(\tilde{C}_{\pm}\pm\varepsilon)(d(x))^{\mu}\big)\end{equation}$$such that for any x∈Ωδ1$ x\in\Omega_{\delta_{1}} $g(w±(d(x)))=g(ξ±ψ(K(d(x))))+ξ±ψ(K(d(x)))g′(Θ±(d(x)))(C±±ε)(d(x))μ.$$\begin{split}g(w_{\pm}(d(x)))&=g(\xi_{\pm}\psi(K(d(x))))+\xi_{\pm}\psi(K(d(x)))g'(\Theta_{\pm}(d(x)))(C_{\pm}\pm\varepsilon)(d(x))^{\mu}.\end{split}$$By Proposition 2.2, we see that (4.2) still holds. Moreover, we can adjust δ1 > 0 such that (4.3) holds here.Proof. Similar to the proof of Theorem 1.1, the proof is divided into the following two steps.Step 1. We first define functions I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as follows.For fixed ε > 0 and ∀x∈Ωδ1$ \forall\,x\in\Omega_{\delta_{1}} $, we defineI 1 ± ( d ( x ) ) = 1 ( d ( x ) ) μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 m ^ ± − g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ; I 2 ± ( d ( x ) ) = C ~ ± ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × N + μ ( N − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + 2 μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + μ × ( μ − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 B ± ∏ i = 1 N − 1 1 − d ( x ) κ i ( x ¯ ) − 1 − ξ ± ψ ( K ( d ( x ) ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) g ′ Θ ± ( d ( x ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ± ε ψ ′ ( x ) ( x ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × N + μ ( N − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + 2 μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + μ × ( μ − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 × m ^ + ∏ i = 1 N − 1 1 − d ( x ) κ i ( x ¯ ) − 1 − ξ ± ψ ( K ( d ( x ) ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) g ′ Θ ± ( d ( x ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) × g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) − B 0 ± ε g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ; I 3 ± ( d ( x ) ) = ( d ( x ) ) − μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × m ^ ± ∑ i = 1 N − 1 C N − 1 i ( − 1 ) i + 1 m ± d ( x ) i 1 − d ( x ) m ± N − 1 − B 0 ± ε C ~ ± ± ε ξ ± ψ ( K ( d ( x ) ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) × g ′ Θ ± ( d ( x ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ( d ( x ) ) μ + ζ ~ ± ( d ( x ) ) ( d ( x ) ) μ ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) k ( x ¯ ) , $$\begin{equation}\begin{array}{l}I_{1 \pm}(d(x))=\frac{1}{(d(x))^{\mu}}\left[\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right) \hat{m}_{\pm}-\frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))}\right]; \\ I_{2 \pm}(d(x))=\tilde{C}_{\pm}\left\{\left[\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\right.\right. \\ \times\left(N+\mu(N-1) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}\right)+\frac{2 \mu \psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}+\mu \\ \left.\times(\mu-1) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}\right] \mathscr{B}_{\pm} \prod\limits_{i=1}^{N-1}\left(1-d(x) \kappa_{i}(\bar{x})\right)^{-1} \\ \left.-\frac{\xi_{\pm} \psi(K(d(x))) g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)}{g\left(\xi_{\pm} \psi(K(d(x)))\right)} \frac{g^{\prime}\left(\Theta_{\pm}(d(x))\right)}{g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)} \frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))}\right\}\\ \pm \varepsilon\left\{\left[\left(\frac{\left.\left.\psi^{\prime}(x)(x)\right)\right)}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\right.\right. \\ \times\left(N+\mu(N-1) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}\right)+\frac{2 \mu \psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}+\mu \\ \left.\times(\mu-1) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}\right] \\ \times \hat{m}_{+} \prod\limits_{i=1}^{N-1}\left(1-d(x) \kappa_{i}(\bar{x})\right)^{-1}-\frac{\xi_{\pm} \psi(K(d(x))) g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)}{g\left(\xi_{\pm} \psi(K(d(x)))\right)} \frac{g^{\prime}\left(\Theta_{\pm}(d(x))\right)}{g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)} \\ \left.\times \frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))}\right\}-\left(B_{0} \pm \varepsilon\right) \frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))};\\ I_{3 \pm}(d(x))=(d(x))^{-\mu}\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right) \\ \times \frac{\hat{m}_{\pm} \sum_{i=1}^{N-1} C_{N-1}^{i}(-1)^{i+1}\left(m_{\pm} d(x)\right)^{i}}{\left(1-d(x) m_{\pm}\right)^{N-1}}-\left(B_{0} \pm \varepsilon\right)\left(\tilde{C}_{\pm} \pm \varepsilon\right) \frac{\xi_{\pm} \psi(K(d(x))) g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)}{g\left(\xi_{\pm} \psi(K(d(x)))\right)} \\ \times \frac{g^{\prime}\left(\Theta_{\pm}(d(x))\right)}{g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)} \frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))}(d(x))^{\mu}+\frac{\tilde{\zeta}_{\pm}(d(x))}{(d(x))^{\mu}} \prod\limits_{i=1}^{N-1} \frac{\kappa_{i}(\bar{x})}{1-d(x)_{k}(\bar{x})},\end{array} \end{equation}$$whereB±=mˆ±,if B0≥0,B±=mˆ∓, if B0<0,$$\begin{equation}\left\{\begin{array}{ll}\mathscr{B}_{\pm}=\hat{m}_{\pm}, & \text {if } B_{0} \geq 0, \\ \mathscr{B}_{\pm}=\hat{m}_{\mp}, & \text { if } B_{0}<0,\end{array}\right. \end{equation}$$m ± are defined by (4.6) andζ ~ ± ( d ( x ) ) = C ~ ± ± ε 2 ( N − 1 ) 1 + ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) × 1 + μ ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) ( d ( x ) ) 2 μ + 2 μ C ~ ± ± ε 2 ( N − 1 ) 1 + μ ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) × K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) ( d ( x ) ) 2 μ + μ ( μ − 1 ) C ~ ± ± ε 2 ( N − 1 ) 1 + μ ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) × K ( d ( x ) ) d ( x ) k ( d ( x ) ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 ( d ( x ) ) 2 μ + v ~ ± ( d ( x ) ) × 1 + ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) 1 + C ~ ± ± ε ( d ( x ) ) μ + 2 μ C ~ ± ± ε ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) ( d ( x ) ) μ + μ ( μ − 1 ) C ~ ± ± ε ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 ( d ( x ) ) μ $$ \begin{equation}\begin{array}{l}\tilde{\zeta}_{\pm}(d(x))=\left(\tilde{C}_{\pm} \pm \varepsilon\right)^{2}(N-1)\left(1+\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}\right) \\ \times\left(1+\frac{\mu \psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}\right)(d(x))^{2 \mu} \\ +2 \mu\left(\tilde{C}_{\pm} \pm \varepsilon\right)^{2}(N-1)\left(1+\frac{\mu \psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}\right) \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \\ \times \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}(d(x))^{2 \mu}+\mu(\mu-1)\left(\tilde{C}_{\pm} \pm \varepsilon\right)^{2}(N-1)\left(1+\frac{\mu \psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))}\right.\\ \left.\times \frac{K(d(x))}{d(x) k(d(x))}\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}(d(x))^{2 \mu}+\tilde{v}_{\pm}(d(x)) \\ \times\left[\left(1+\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}\right)\left(1+\left(\tilde{C}_{\pm} \pm \varepsilon\right)(d(x))^{\mu}\right)\right. \\ +2 \mu\left(\tilde{C}_{\pm} \pm \varepsilon\right) \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}(d(x))^{\mu} \\ \left.+\mu(\mu-1)\left(\tilde{C}_{\pm} \pm \varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}(d(x))^{\mu}\right]\end{array} \end{equation}$$withν˜±(d(x))=∑i=2N−1CN−1i(R˜±(d(x)))i,$$\tilde{\nu}_{\pm}(d(x))=\sum_{i=2}^{N-1}C_{N-1}^{i}(\tilde{\mathfrak{R}}_{\pm}(d(x)))^{i},$$where CN−1i$ C_{N-1}^{i} $is given in (4.7) andR˜±(d(x))=(C˜±±ε)(1+μψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x)))(d(x))μ.$$\tilde{\mathfrak{R}}_{\pm}(d(x))=(\tilde{C}_{\pm}\pm\varepsilon)\bigg(1+\frac{\mu\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}\bigg)(d(x))^{\mu}.$$Next, we prove(4.17)limd(x)→0(I1±(d(x))+I2±(d(x))+I3±(d(x)))=±εϱ±mˆ±,$$\begin{equation}\lim_{d(x)\rightarrow0}(I_{1\pm}(d(x))+I_{2\pm}(d(x))+I_{3\pm}(d(x)))=\pm\varepsilon\varrho_{\pm}\hat{m}_{\pm},\end{equation}$$where(4.18)ϱ±=(1−μ)(γ+N)Dk−(N+1)γ−1N+μ(N−1)γ+NN+1Dk+μγ+Nγ−1DkN+μ(N−1)γ+NN+1Dk−2+μ(1−μ)(γ+N)2(γ−1)(N+1)Dk2mˆ+mˆ±+(γ+N)Dk−(N+1).$$\begin{equation}\begin{array}{l}\varrho_{\pm}=\left[\frac{(1-\mu)(\gamma+N) D_{k}-(N+1)}{\gamma-1}\left(N+\mu(N-1) \frac{\gamma+N}{N+1} D_{k}\right)\right. \\ \left.+\mu \frac{\gamma+N}{\gamma-1} D_{k}\left(N+\mu(N-1) \frac{\gamma+N}{N+1} D_{k}-2\right)+\mu(1-\mu) \frac{(\gamma+N)^{2}}{(\gamma-1)(N+1)} D_{k}^{2}\right] \frac{\hat{m}_{+}}{\hat{m}_{\pm}} \\ +(\gamma+N) D_{k}-(N+1).\end{array} \end{equation}$$To prove (4.17), we calculate the limits of I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as d(x)→0.$d(x) \rightarrow 0.$• First, we investigate the limits of I1±(d(x)) as d(x)→0.$I_{1 \pm}(d(x)) \text { as } d(x) \rightarrow 0.$As the preliminaries, we prove some limits as below.Case 1. When Dk ∈ (0, 1), by a direct calculation, we haveN+1(1−μ)Dk−N>N+1(1−μ)−N>(1+μ)N+1N+1−μ.$$\frac{N+1}{(1-\mu)D_{k}}-N>\frac{N+1}{(1-\mu)}-N>\frac{(1+\mu)N+1}{N+1-\mu}.$$So, by (1.15), Lemma 3.1 (ii) and Lemma 3.6 (viii) and (x) with ρ=μ ∈ (0, 1), we obtain(4.19)limd(x)→0+(d(x))−μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1K(d(x))k′(d(x))k2(d(x))mˆ±=limd(x)→0+K(d(x))d(x)μ(K(d(x)))−μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1K(d(x))k′(d(x))k2(d(x))mˆ±=0, if S2 and g3 hold with θ(N+1)μ−N>γ>N+1(1−μ)Dk−N,0, if s1 holds with γ>N+1(1−μ)Dk−N in g2$$\begin{equation}\begin{aligned} & \lim _{d(x) \rightarrow 0^{+}}(d(x))^{-\mu}\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}+\frac{\gamma+N}{\gamma-1}\right) \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))} \hat{m}_{\pm} \\=& \lim _{d(x) \rightarrow 0^{+}}\left(\frac{K(d(x))}{d(x)}\right)^{\mu}(K(d(x)))^{-\mu}\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}+\frac{\gamma+N}{\gamma-1}\right) \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))} \hat{m}_{\pm} \\=&\left\{\begin{array}{ll}0, & \text { if }\left(\mathbf{S}_{2}\right) \text { and }\left(\mathbf{g}_{3}\right) \text { hold with } \frac{\theta(N+1)}{\mu}-N>\gamma>\frac{N+1}{(1-\mu) D_{k}}-N, \\ 0, & \text { if }\left(\mathbf{s}_{1}\right) \text { holds with } \gamma>\frac{N+1}{(1-\mu) D_{k}}-N \text { in }\left(\mathbf{g}_{2}\right)\end{array}\right.\end{aligned} \end{equation}$$and(4.20)limd(x)→0(d(x))−μ(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=limd(x)→0(K(d(x))d(x))μ(K(d(x)))−μ(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=0, if (S2) and (g3) hold with θ(N+1)μ−N>γ,0, if (S1) holds in (g2).$$\begin{equation}\begin{split}&\lim\limits_{d(x)\rightarrow0}(d(x))^{-\mu}\bigg(\xi_{\pm}^{-(\gamma+N)}-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg)\\=&\lim\limits_{d(x)\rightarrow0}\bigg(\frac{K(d(x))}{d(x)}\bigg)^{\mu}(K(d(x)))^{-\mu}\bigg(\xi_{\pm}^{-(\gamma+N)}-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg)\\=&\begin{cases} 0, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and}\mathbf{(g_{3})} \mbox{ hold with }\frac{\theta(N+1)}{\mu}-N>\gamma,\\0, &\mbox{ if }\mathbf{(S_{1})}\mbox{ holds} \mbox{ in}\mathbf{(g_{2})}.\end{cases}\end{split}\end{equation}$$Case 2. When Dk ∈ [1, ∞), by (1.15), Lemma 3.1 (ii) and Lemma 3.6 (viii) and (x) with ρ=1, we obtain(4.21)limd(x)→0(d(x))−μ(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1)K(d(x))k′(d(x))k2(d(x))mˆ±=limd(x)→0K(d(x))(d(x))μ(K(d(x)))−1(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1)K(d(x))k′(d(x))k2(d(x))mˆ±=0, if (S2) and (g3) hold with θ(N+1)−N>γ>2N+1N,0, if (S1) holds with γ>2N+1N in (g2)$$\begin{equation}\begin{split}&\lim\limits_{d(x)\rightarrow0}(d(x))^{-\mu}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}+\frac{\gamma+N}{\gamma-1}\bigg)\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}\hat{m}_{\pm}\\=&\lim\limits_{d(x)\rightarrow0}\frac{K(d(x))}{(d(x))^{\mu}}(K(d(x)))^{-1}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}+\frac{\gamma+N}{\gamma-1}\bigg)\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}\hat{m}_{\pm}\\=&\begin{cases} 0, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and}\mathbf{(g_{3})} \mbox{ hold with }\theta(N+1)-N>\gamma>\frac{2N+1}{N},\\0, &\mbox{ if }\mathbf{(S_{1})}\mbox{ holds with }\gamma>\frac{2N+1}{N} \mbox{ in }\mathbf{(g_{2})}\end{cases}\end{split}\end{equation}$$and(4.22)limd(x)→0(d(x))−μ(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=limd(x)→0K(d(x))(d(x))μ(K(d(x)))−1(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=0, if (S2) and (g3) hold with θ(N+1)−N>γ,0, if (S1) holds in (g2).$$\begin{equation}\begin{split}&\lim\limits_{d(x)\rightarrow0}(d(x))^{-\mu}\bigg(\xi_{\pm}^{-(\gamma+N)}-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg)\\=&\lim\limits_{d(x)\rightarrow0}\frac{K(d(x))}{(d(x))^{\mu}}(K(d(x)))^{-1}\bigg(\xi_{\pm}^{-(\gamma+N)}-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg)\\=&\begin{cases} 0, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and}\mathbf{(g_{3})} \mbox{ hold with }\theta(N+1)-N>\gamma,\\0, &\mbox{ if }\mathbf{(S_{1})}\mbox{ holds} \mbox{ in}\mathbf{(g_{2})}.\end{cases}\end{split}\end{equation}$$On the other hand, a simple calculation shows that(4.23)−limd(x)→0γ+Nγ−1(d(x))−μ(K(d(x))k′(d(x))k2(d(x))−(1−Dk))mˆ±=−limd(x)→0γ+Nγ−1(d(x))1−μ(d(x))−1(K(d(x))k′(d(x))k2(d(x))−(1−Dk))mˆ±=0$$\begin{equation}\begin{split}-&\lim\limits_{d(x)\rightarrow0}\frac{\gamma+N}{\gamma-1}(d(x))^{-\mu}\bigg(\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}-(1-D_{k})\bigg)\hat{m}_{\pm}\\=-&\lim\limits_{d(x)\rightarrow0}\frac{\gamma+N}{\gamma-1}(d(x))^{1-\mu}(d(x))^{-1}\bigg(\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}-(1-D_{k})\bigg)\hat{m}_{\pm}=0\end{split}\end{equation}$$We obtain by combining (4.19)-(4.20) (or (4.21)-(4.22)) and (4.23) that(4.24)limd(x)→0I1±(d(x))=0.$$\begin{equation}\lim\limits_{d(x)\rightarrow0}I_{1\pm}(d(x))=0.\end{equation}$$• Second, by (4.2), Lemma 3.1 (ii), Lemma 3.6 (iii)-(iv) and the choices of ξ ± , C˜±$ \tilde{C}_{\pm} $in Theorem 1.2, we obtain(4.25)limd(x)→0I2±(d(x))=C˜±[((1−μ)(γ+N)Dk−(N+1)γ−1(N+μ(N−1)γ+NN+1Dk)+μγ+Nγ−1Dk(N+μ(N−1)γ+NN+1Dk−2)+μ(1−μ)(γ+N)2(γ−1)(N+1)Dk2)B±+γ((γ+N)Dk−(N+1))mˆ±γ−1]−B0((γ+N)Dk−(N+1))mˆ±γ−1±εϱ±mˆ±=±εϱ±mˆ±, where ϱ± are given by (4.18).$$\begin{equation}\begin{split}&\lim\limits_{d(x)\rightarrow0}I_{2\pm}(d(x))=\tilde{C}_{\pm}\bigg[\bigg(\frac{(1-\mu)(\gamma+N)D_{k}-(N+1)}{\gamma-1}\bigg(N+\mu(N-1)\frac{\gamma+N}{N+1}D_{k}\bigg)\\&+\mu\frac{\gamma+N}{\gamma-1}D_{k}\bigg(N+\mu(N-1)\frac{\gamma+N}{N+1}D_{k}-2\bigg)+\mu(1-\mu)\frac{(\gamma+N)^{2}}{(\gamma-1)(N+1)}D_{k}^{2}\bigg)\mathscr{B}_{\pm}\\&+\frac{\gamma\big((\gamma+N)D_{k}-(N+1)\big)\hat{m}_{\pm}}{\gamma-1}\bigg]-B_{0}\frac{\big((\gamma+N)D_{k}-(N+1)\big)\hat{m}_{\pm}}{\gamma-1}\\&\pm\varepsilon\varrho_{\pm}\hat{m}_{\pm}=\pm\varepsilon\varrho_{\pm} \hat{m}_{\pm},\mbox{ where} \varrho_{\pm} \mbox{ are given by } (4.18).\end{split}\end{equation}$$• Third, we conclude by Lemma 3.1 (ii) Lemma 3.6 (iii)-(iv) and a straightforward calculation that(4.26)limd(x)→0I3±(d(x))=0.$$\begin{equation}\lim\limits_{d(x)\rightarrow0}I_{3\pm}(d(x))=0.\end{equation}$$Combining (4.24) and (4.25)-(4.26), we obtain (4.17) holds. Moreover, it follows by (1.15) that thatϱ ± >(γ+N)Dk−(N + 1)>0. By (b1)-(b2) and (4.17), we see that there exists a sufficiently small constant δε > 0 such that (4.12) and (4.4)-(4.5) hold here.Step 2. Letu_ε(x)=−ξ+ψ(K(d(x)))(1+(C˜++ε)(d(x))μ),x∈Ωδε.$$\underline{u}_{\varepsilon}(x)=-\xi_{+}\psi(K(d(x)))\big(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}\big),\,x\in\Omega_{\delta_{\varepsilon}}.$$Theng(−u_ε(x))=g(ξ+ψ(K(d(x))))+ξ+ψ(K(d(x)))g′(Θ+(d(x)))(C˜++ε)(d(x))μ,x∈Ωδε,$$g(-\underline{u}_{\varepsilon}(x))=g(\xi_{+}\psi(K(d(x))))+\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu},\,x\in\Omega_{\delta_{\varepsilon}},$$where Θ+(d(x)) is given by (4.16). By Lemma 4.2, we have for any x∈Ωδε(4.27)det(D2u_ε(x))−b(x)g(−u_ε(x))≥−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))×(1+(C˜++ε)(N−1)(d(x))μ+μ(C˜++ε)(N−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ+ν˜+(d(x)))[(1+ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x)))(1+(C˜++ε)(d(x))μ)+2μ(C++ε)ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ+μ(μ−1)(C˜++ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2(d(x))μ]×∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)−kN+1(d(x))(1+(B0+ε)(d(x))μ)[g(ξ+ψ(K(d(x))))+ξ+ψ(K(d(x)))g′(Θ+(d(x)))(C˜++ε)(d(x))μ]=−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ×{(d(x))−μ(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)+(C˜++ε)[(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(N+μ(N−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))×K(d(x))d(x)k(d(x)))+2μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))+μ(μ−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))×ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2]∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)+ζ˜+(d(x))(d(x))μ∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)}−kN+1(d(x))g(ξ+ψ(K(d(x))))−(B0+ε)kN+1(d(x))g(ξ+ψ(K(d(x))))(d(x))μ−(C˜++ε)kN+1(d(x))ξ+ψ(K(d(x)))g′(Θ+(d(x)))(d(x))μ−(B0+ε)(C˜++ε)kN+1(d(x))×ξ+ψ(K(d(x)))g′(Θ+(d(x)))(d(x))2μ=−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ×{(d(x))−μ[(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ+−g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))]+(C˜++ε)[((ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(N+μ(N−1)×ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x)))+2μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))+μ(μ−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2)∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)−ξ+ψ(K(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))g′(Θ+(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))]−(B0+ε)g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))−(B0+ε)(C˜++ε)ξ+ψ(K(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))×g′(Θ+(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))(d(x))μ+(d(x))−μ(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ+∑i=1N−1CN−1i(−1)i+1(m+d(x))i(1−d(x)m+)N−1+ζ˜+(d(x))(d(x))μ∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)}≥−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ∑i=13Ii+(d(x))>0,$$\begin{align}\nonumber&\mbox{det}(D^{2}\underline{u}_{\varepsilon}(x))-b(x)g(-\underline{u}_{\varepsilon}(x))\geq-\xi_{+}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))\\\nonumber&\times\bigg(1+(\tilde{C}_{+}+\varepsilon)(N-1)(d(x))^{\mu}+\mu(\tilde{C}_{+}+\varepsilon)(N-1)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}(d(x))^{\mu}\\\nonumber&+\tilde{\nu}_{+}(d(x))\bigg)\bigg[\bigg(1+\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}\bigg)\big(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}\big)\\\nonumber&+2\mu(C_{+}+\varepsilon)\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}(d(x))^{\mu}\\\nonumber&+\mu(\mu-1)(\tilde{C}_{+}+\varepsilon)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}(d(x))^{\mu}\bigg]\\\nonumber&\times\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}-k^{N+1}(d(x))\big(1+(B_{0}+\varepsilon)(d(x))^{\mu}\big)\big[g(\xi_{+}\psi(K(d(x))))\\\nonumber&+\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}\big]\\\nonumber&=-\xi_{+}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))(d(x))^{\mu}\\\nonumber&\times\bigg\{(d(x))^{-\mu}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\\\nonumber&+(\tilde{C}_{+}+\varepsilon)\bigg[\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\bigg(N+\mu(N-1)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\\\nonumber&\times\frac{K(d(x))}{d(x)k(d(x))}\bigg)+\frac{2\mu\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}+\frac{\mu(\mu-1)\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\\\nonumber&\times\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}\bigg]\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}+\frac{\tilde{\zeta}_{+}(d(x))}{(d(x))^{\mu}}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\bigg\}\\\nonumber&-k^{N+1}(d(x))g(\xi_{+}\psi(K(d(x))))-(B_{0}+\varepsilon)k^{N+1}(d(x))g(\xi_{+}\psi(K(d(x))))(d(x))^{\mu}\\\nonumber&-(\tilde{C}_{+}+\varepsilon)k^{N+1}(d(x))\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(d(x))^{\mu}-(B_{0}+\varepsilon)(\tilde{C}_{+}+\varepsilon)k^{N+1}(d(x))\\\nonumber&\times\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(d(x))^{2\mu}\\\nonumber&=-\xi_{+}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))(d(x))^{\mu}\\\nonumber&\times\bigg\{(d(x))^{-\mu}\bigg[\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\hat{m}_{+}-\frac{g(\xi_{+}\psi(K(d(x))))}{\xi_{+}^{N}g(\psi(K(d(x))))}\bigg]\\\nonumber&+(\tilde{C}_{+}+\varepsilon)\bigg[\bigg(\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\bigg(N+\mu(N-1)\\\label{4.27l}&\times\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}\bigg)+\frac{2\mu\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}\\\nonumber&+\frac{\mu(\mu-1)\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}\bigg)\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\\\nonumber&-\frac{\xi_{+}\psi(K(d(x)))g'(\xi_{+}\psi(K(d(x))))}{g(\xi_{+}\psi(K(d(x))))}\frac{g'(\Theta_{+}(d(x)))}{g'(\xi_{+}\psi(K(d(x))))}\frac{g(\xi_{+}\psi(K(d(x))))}{\xi_{+}^{N}g(\psi(K(d(x))))}\bigg]\\\nonumber&-(B_{0}+\varepsilon)\frac{g(\xi_{+}\psi(K(d(x))))}{\xi_{+}^{N}g(\psi(K(d(x))))}-(B_{0}+\varepsilon)(\tilde{C}_{+}+\varepsilon)\frac{\xi_{+}\psi(K(d(x)))g'(\xi_{+}\psi(K(d(x))))}{g(\xi_{+}\psi(K(d(x))))}\\\nonumber&\times\frac{g'(\Theta_{+}(d(x)))}{g'(\xi_{+}\psi(K(d(x))))}\frac{g(\xi_{+}\psi(K(d(x))))}{\xi_{+}^{N}g(\psi(K(d(x))))}(d(x))^{\mu}\\\nonumber&+(d(x))^{-\mu}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\frac{\hat{m}_{+}\sum\limits_{i=1}^{N-1}C_{N-1}^{i}(-1)^{i+1}(m_{+}d(x))^{i}}{(1-d(x)m_{+})^{N-1}}\\\nonumber&+\frac{\tilde{\zeta}_{+}(d(x))}{(d(x))^{\mu}}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\bigg\}\\&\geq-\xi_{+}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))(d(x))^{\mu}\sum\limits_{i=1}^{3}I_{i+}(d(x))>0, \nonumber\end{align}$$i.e., uε is a subsolution of Eq. (1.1) in Ωδε. Moreover, it follows by Lemma 4.3 that for i=1, ... ,NSi(D2u_ε(x))=ξ+i(ψ′(K(d(x))))iki(d(x))(1+(C˜++ε)(d(x))μ+μ(C˜++ε)K(d(x))d(x)k(d(x))×ψ(K(d(x)))ψ′(K(d(x)))K(d(x))(d(x))μ)iSi(ϵ1,⋅⋅⋅,ϵN−1)−ξ+i(ψ′(K(d(x))))i−1ψ′′(K(d(x)))ki+1(d(x))(1+(C˜++ε)(d(x))μ+μ(C˜++ε)×ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ)i−1[(1+ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x)))×(1+(C˜++ε)(d(x))μ)+2μ(C++ε)ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ+μ(μ−1)(C˜++ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2(d(x))μ]×Si−1(ϵ1,⋅⋅⋅,ϵN−1).$$\begin{split}&S_{i}(D^{2}\underline{u}_{\varepsilon}(x))=\xi_{+}^{i}(\psi'(K(d(x))))^{i}k^{i}(d(x))\bigg(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}+\mu(\tilde{C}_{+}+\varepsilon)\frac{K(d(x))}{d(x)k(d(x))}\\&\times\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}(d(x))^{\mu}\bigg)^{i}S_{i}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1})\\&-\xi_{+}^{i}(\psi'(K(d(x))))^{i-1}\psi''(K(d(x)))k^{i+1}(d(x))\bigg(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}+\mu(\tilde{C}_{+}+\varepsilon)\\&\times\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}(d(x))^{\mu}\bigg)^{i-1}\bigg[\bigg(1+\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}\bigg)\\&\times\big(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}\big)+2\mu(C_{+}+\varepsilon)\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}(d(x))^{\mu}\\&+\mu(\mu-1)(\tilde{C}_{+}+\varepsilon)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}(d(x))^{\mu}\bigg]\\&\timesS_{i-1}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1}).\end{split}\quad\quad\quad\quad\quad\,$$This implies that we can adjust the above positive constant δε such that for any x∈ΩδεS i ( D 2 u _ ε ( x ) ) > 0 for i = 1 , ⋅ ⋅ ⋅ , N . $$S_{i}(D^{2}\underline{u}_{\varepsilon}(x))>0 \,\mbox{for}\,i=1,\cdot\cdot\cdot,N.$$We obtain by Lemma 4.4 that D2uε is positive definite in Ωδε.Letu‾ε(x)=−ξ−ψ(K(d(x)))(1+(C˜−−ε)(d(x))μ),x∈Ωδε$$\begin{split}&\overline{u}_{\varepsilon}(x)=-\xi_{-}\psi(K(d(x)))\big(1+(\tilde{C}_{-}-\varepsilon)(d(x))^{\mu}\big),\,x\in\Omega_{\delta_{\varepsilon}}\end{split}$$By the same calculation as (4.27), we obtaindet(D2u‾ε(x))−b(x)g(−u‾ε(x))≤−ξ−N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ∑i=13Ii−(d(x))<0,$$\begin{split}&\mbox{det}(D^{2}\overline{u}_{\varepsilon}(x))-b(x)g(-\overline{u}_{\varepsilon}(x))\\&\leq-\xi_{-}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))(d(x))^{\mu}\sum_{i=1}^{3}I_{i-}(d(x))<0,\end{split}$$i.e. uε is a supersolution of Eq. (1.1) in Ωδε.Let u be the unique strictly convex solution to problem (1.1). Through the same argument as Theorem 1.1, we see that there exists a large constant M > 0 such thatu_ε(x)−Md(x)≤u(x)≤u‾ε(x)+Md(x),x∈Ωδε.$$\underline{u}_{\varepsilon}(x)-Md(x)\leq u(x)\leq\overline{u}_{\varepsilon}(x)+Md(x),\,x\in\Omega_{\delta_{\varepsilon}}.$$Hence, for any x∈ ΩδεC˜++ε+M(d(x))1−μξ+ψ(K(d(x)))≥(−u(x)ξ+ψ(K(d(x)))−1)(d(x))−μ;C˜−−ε−M(d(x))1−μξ−ψ(K(d(x)))≤(−u(x)ξ−ψ(K(d(x)))−1)(d(x))−μ.$$\begin{split}\tilde{C}_{+}+\varepsilon+\frac{M(d(x))^{1-\mu}}{\xi_{+}\psi(K(d(x)))}&\geq\bigg(\frac{-u(x)}{\xi_{+}\psi(K(d(x)))}-1\bigg)(d(x))^{-\mu};\\\tilde{C}_{-}-\varepsilon-\frac{M(d(x))^{1-\mu}}{\xi_{-}\psi(K(d(x)))}&\leq\bigg(\frac{-u(x)}{\xi_{-}\psi(K(d(x)))}-1\bigg)(d(x))^{-\mu}.\end{split}$$Since (1-μ)(γ+N)Dk-(N+1)>0, we conclude from Lemma 3.1 (ii) , Lemma 3.6 (iii), Proposition 2.7 and Proposition 2.5 (ii) thatC˜++ε≥lim supd(x)→0(−u(x)ξ+ψ(K(d(x)))−1)(d(x))−μ;C˜−−ε≤lim infd(x)→0(−u(x)ξ−ψ(K(d(x)))−1)(d(x))−μ.$$\begin{split}\tilde{C}_{+}+\varepsilon&\geq\limsup_{d(x)\rightarrow0}\bigg(\frac{-u(x)}{\xi_{+}\psi(K(d(x)))}-1\bigg)(d(x))^{-\mu};\\\tilde{C}_{-}-\varepsilon&\leq\liminf_{d(x)\rightarrow0}\bigg(\frac{-u(x)}{\xi_{-}\psi(K(d(x)))}-1\bigg)(d(x))^{-\mu}.\end{split}$$Letting ε → 0, the proof is finished. □4.3Proof of Theorem 1.3Now, we prove Theorem 1.3. As before, for fixed ε > 0, we definew ± ( d ( x ) ) = η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + ( C ± ∗ ± ε ) M ( d ( x ) ) ) , x ∈ Ω δ 1 , $$w_{\pm}(d(x))=\eta_{\pm}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x))\big),\,x\in\Omega_{\delta_{1}},$$where η ± , C ± * and M$ \mathfrak{M} $are given in Theorem 1.3. By the Lagrange’s mean value theorem, it is clear that there exist λ ± ∈(0, 1) and(4.28)Θ ± ( d ( x ) ) = η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + λ ± ( C ± ∗ ± ε ) M ( d ( x ) ) ) $$\begin{equation}\Theta_{\pm}(d(x))=\eta_{\pm}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+\lambda_{\pm}(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x))\big)\end{equation}$$such that for x∈Ωδ1g ( w ± ( d ( x ) ) ) = g ( η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ) + η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) g ′ ( Θ ± ( d ( x ) ) ) ( C ± ∗ ± ε ) M ( d ( x ) ) . $$\begin{split}g(w_{\pm}(d(x)))&=g\bigg(\eta_{\pm}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\bigg)\\&+\eta_{\pm}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)g'(\Theta_{\pm}(d(x)))(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x)).\end{split}$$We obtain by Proposition 2.2 that (4.2) still holds. Moreover, we can adjust δ1 such that (4.3) holds here.Proof. The proof is still divided into the following two steps.Step 1. As before, for fixed ε> 0 and ∀ x∈Ωδ1, we define(4.29)r ( d ( x ) ) = ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 $$\begin{equation}r(d(x))=\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\end{equation}$$andI 1 ± ( d ( x ) ) = ( M ( d ( x ) ) ) − 1 { [ ( N N + 1 ) N ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] m ^ ± − g ( ξ ± ψ ( r ( d ( x ) ) ) ) ξ ± N g ( ψ ( r ( d ( x ) ) ) ) } ; I 2 ± ( d ( x ) ) = C ± ∗ [ ( N N + 1 ) N N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) C ± ∏ i = 1 N − 1 ( 1 − d ( x ) κ i ( x ¯ ) ) − 1 − η ± ψ ( r ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) g ′ ( Θ ± ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ] ± ε [ ( N N + 1 ) N N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) m ^ + ∏ i = 1 N − 1 ( 1 − d ( x ) κ i ( x ¯ ) ) − 1 − η ± ψ ( r ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) g ′ ( Θ ± ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ] ; I 3 ± ( d ( x ) ) = − ( B 0 ± ε ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ ( M ( d ( x ) ) ) − 1 + ( M ( d ( x ) ) ) − 1 [ ( N N + 1 ) N ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] m ^ ± ∑ i = 1 N − 1 C N − 1 i ( − 1 ) i + 1 ( m ± d ( x ) ) i ( 1 − d ( x ) m ± ) N − 1 − ( B 0 ± ε ) ( C ± ∗ ± ε ) η ± ψ ( r ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) g ′ ( Θ ± ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ + ( N N + 1 ) N ζ ± ∗ ( d ( x ) ) ( M ( d ( x ) ) ) − 1 ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) , $$\begin{align}&I_{1\pm}(d(x))=(\mathfrak{M}(d(x)))^{-1}\bigg\{\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\ &+\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]\hat{m}_{\pm}-\frac{g(\xi_{\pm}\psi(r(d(x))))}{\xi_{\pm}^{N}g(\psi(r(d(x))))}\bigg\};\\ &I_{2\pm}(d(x))=C_{\pm}^{*}\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}N\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\mathscr{C}_{\pm}\prod\limits_{i=1}^{N-1}(1-d(x)\kappa_{i}(\bar{x}))^{-1}\\ &-\frac{\eta_{\pm}\psi(r(d(x)))g'(\eta_{\pm}\psi(r(d(x))))}{g(\eta_{\pm}\psi(r(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\eta_{\pm}\psi(r(d(x))))}\frac{g(\eta_{\pm}\psi(r(d(x))))}{\eta_{\pm}^{N}g(\psi(r(d(x))))}\bigg]\\ &\pm\varepsilon\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}N\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\hat{m}_{+}\prod\limits_{i=1}^{N-1}(1-d(x)\kappa_{i}(\bar{x}))^{-1}\\ &-\frac{\eta_{\pm}\psi(r(d(x)))g'(\eta_{\pm}\psi(r(d(x))))}{g(\eta_{\pm}\psi(r(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\eta_{\pm}\psi(r(d(x))))}\frac{g(\eta_{\pm}\psi(r(d(x))))}{\eta_{\pm}^{N}g(\psi(r(d(x))))}\bigg];\\ &I_{3\pm}(d(x))=-(B_{0}\pm\varepsilon)\frac{g(\eta_{\pm}\psi(r(d(x))))}{\eta_{\pm}^{N}g(\psi(r(d(x))))}(d(x))^{\mu}(\mathfrak{M}(d(x)))^{-1}\\ &+(\mathfrak{M}(d(x)))^{-1}\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\ &+\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]\frac{\hat{m}_{\pm}\sum_{i=1}^{N-1}C_{N-1}^{i}(-1)^{i+1}(m_{\pm}d(x))^{i}}{(1-d(x)m_{\pm})^{N-1}}\\ &-(B_{0}\pm\varepsilon)(C_{\pm}^{*}\pm\varepsilon)\frac{\eta_{\pm}\psi(r(d(x)))g'(\eta_{\pm}\psi(r(d(x))))}{g(\eta_{\pm}\psi(r(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\eta_{\pm}\psi(r(d(x))))}\frac{g(\eta_{\pm}\psi(r(d(x))))}{\eta_{\pm}^{N}g(\psi(r(d(x))))}(d(x))^{\mu}\\ &+\bigg(\frac{N}{N+1}\bigg)^{N}\zeta_{\pm}^{*}(d(x))(\mathfrak{M}(d(x)))^{-1}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})},\end{align}$$whereif(S2) holds and θ=0,C±=mˆ±, if X+≥0 and X−≥0,C∓=mˆ∓, if X+≤0 and X−≤0,C+=C−=mˆ+, if X+>0 and X−<0,C+=C−=mˆ−, if X+<0 and X−>0; if (S1)holds,C±=mˆ±,$$\begin{cases}\mbox{if}\mathbf{(S_{2})} \mbox{ holds and }\theta=0,\,&\begin{cases} \mathscr{C}_{\pm}=\hat{m}_{\pm},\,&\mbox{ if }\mathfrak{X}_{+}\geq0 \mbox{ and }\mathfrak{X}_{-}\geq0,\\\mathscr{C}_{\mp}=\hat{m}_{\mp},\,&\mbox{ if }\mathfrak{X}_{+}\leq0 \mbox{ and }\mathfrak{X}_{-}\leq0,\\\mathscr{C}_{+}=\mathscr{C}_{-}=\hat{m}_{+},\,&\mbox{ if }\mathfrak{X}_{+}>0 \mbox{ and }\mathfrak{X}_{-}<0,\\\mathscr{C}_{+}=\mathscr{C}_{-}=\hat{m}_{-},\,&\mbox{ if }\mathfrak{X}_{+}<0 \mbox{ and }\mathfrak{X}_{-}>0;\end{cases}\\\mbox{ if }\mathbf{(S_{1})} \mbox{holds},\,\mathscr{C}_{\pm}=\hat{m}_{\pm},\end{cases}$$m ± are defined by (4.6) andζ ± ∗ = ( C ± ∗ ± ε ) ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) M ( d ( x ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( N + ( C ± ∗ ± ε ) ( N − 1 ) M ( d ( x ) ) ) + ( C ± ∗ ± ε ) 2 ( N − 1 ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ′ ( d ( x ) ) d ( x ) L ( d ( x ) ) − 1 ) × ( M ( d ( x ) ) ) 2 + [ β N N + 1 ( C ± ∗ ± ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( M ( d ( x ) ) ) β + 1 β + β N N + 1 ( C ± ± ε ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s ( M ( d ( x ) ) ) β + 1 β + β ( β + 1 ) N ( C ± ± ε ) N + 1 × ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s ( M ( d ( x ) ) ) β + 2 β + β ( C ± ± ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + L ~ ′ ( d ( x ) ) d ( x ) L ( d ( x ) ) − 1 ) ( M ( d ( x ) ) ) β + 1 β ] × ( 1 + ( C ± ∗ ± ε ) ( N − 1 ) M ( d ( x ) ) + ν ± ∗ ( d ( x ) ) ) + ν ± ∗ ( d ( x ) ) [ ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) ( 1 + ( C ± ∗ ± ε ) M ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ( 1 + ( C ± ± ε ) M ( d ( x ) ) ) ] $$ \begin{split} &\zeta_{\pm}^{*}=(C^{*}_{\pm}\pm\varepsilon)\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\mathfrak{M}(d(x))\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds} \\ &\times\big(N+(C^{*}_{\pm}\pm\varepsilon)(N-1)\mathfrak{M}(d(x))\big)+(C^{*}_{\pm}\pm\varepsilon)^{2}(N-1)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{L'(d(x))d(x)}{L(d(x))}-1\bigg) \\ &\times(\mathfrak{M}(d(x)))^{2}+\bigg[\frac{\beta N}{N+1}(C^{*}_{\pm}\pm\varepsilon)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds} \\ &\times(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}+\frac{\beta N}{N+1}(C_{\pm}\pm\varepsilon)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}+\frac{\beta(\beta+1)N(C_{\pm}\pm\varepsilon)}{N+1} \\ &\times\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}(\mathfrak{M}(d(x)))^{\frac{\beta+2}{\beta}} \\ &+\beta(C_{\pm}\pm\varepsilon)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\tilde{L}'(d(x))d(x)}{L(d(x))}-1\bigg)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\bigg] \\ &\times\big(1+(C_{\pm}^{*}\pm\varepsilon)(N-1)\mathfrak{M}(d(x))+\nu_{\pm}^{*}(d(x))\big) \\ &+\nu_{\pm}^{*}(d(x))\bigg[\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\big(1+(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x))\big)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds} \\ &+\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\big(1+(C_{\pm}\pm\varepsilon)\mathfrak{M}(d(x))\big)\bigg] \end{split} $$withν±∗(d(x))=R2±∗(d(x))+∑i=2N−1CN−1i(R1±∗(d(x))+R2±∗(d(x)))i,$$\nu_{\pm}^{*}(d(x))=\mathfrak{R}_{2\pm}^{*}(d(x))+\sum_{i=2}^{N-1}C_{N-1}^{i}\big(\mathfrak{R}_{1\pm}^{*}(d(x))+\mathfrak{R}_{2\pm}^{*}(d(x))\big)^{i},$$where CiN-1 is given in (4.7) andR1±∗(d(x))=(C±∗±ε)M(d(x))andR2±∗(d(x))=β(C±∗±ε)(M(d(x)))β+1β.$$\mathfrak{R}_{1\pm}^{*}(d(x))=(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x))\mbox{ and }\mathfrak{R}_{2\pm}^{*}(d(x))=\beta(C_{\pm}^{*}\pm\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}.$$Next, we prove(4.30)limd(x)→0(I1±(d(x))+I2±(d(x))+I3±(d(x)))=±ε(γ+Nmˆ+mˆ±)η±−(γ+N).$$\begin{equation}\lim_{d(x)\rightarrow0}(I_{1\pm}(d(x))+I_{2\pm}(d(x))+I_{3\pm}(d(x)))=\pm\varepsilon\bigg(\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{\pm}}\bigg)\eta_{\pm}^{-(\gamma+N)}.\end{equation}$$To prove (4.30), we calculate the limits of I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as d(x)→0.∙ First, by Lemma 3.2 (ii)−(iii), Lemma 3.6 (iv), (vi)−(vii) and (ix), we obtain(4.31)limd(x)→0I1±(d(x))=−NM+1N(N+1)σmˆ±(γ−1)2+σlnη±η±γN, if S2 and g3−g4 hold with θ=0,0, if S1 holds ing2.$$\begin{equation}\lim _{d(x) \rightarrow 0} I_{1 \pm}(d(x))=\left\{\begin{array}{ll}-\frac{\left(\frac{N}{M+1}\right)^{N}(N+1) \sigma \hat{m}_{\pm}}{(\gamma-1)^{2}}+\frac{\sigma \ln \eta_{\pm}}{\eta_{\pm}^{\gamma} N}, & \text { if }\left(\mathbf{S}_{2}\right) \text { and }\left(\mathbf{g}_{3}\right)-\left(\mathbf{g}_{4}\right) \text { hold with } \theta=0, \\ 0, & \text { if }\left(\mathbf{S}_{1}\right) \text { holds } \mathrm{in}\left(\mathbf{g}_{2}\right).\end{array}\right.\end{equation} $$∙ Second, by (4.2), Lemma 3.6 (iv) and the choices of η ± and C* ± in Theorem 1.3, we obtain(4.32)limd(x)→0I2±(d(x))=C±∗[(NN+1)NN(γ+N)γ−1C±+γη±−(γ+N)]±ε(γ+Nmˆ+mˆ±)η±−(γ+N).$$\begin{equation}\begin{split}&\lim_{d(x)\rightarrow0}I_{2\pm}(d(x))=C_{\pm}^{*}\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}\frac{N(\gamma+N)}{\gamma-1}\mathscr{C}_{\pm}+\gamma\eta_{\pm}^{-(\gamma+N)}\bigg]\pm\varepsilon\bigg(\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{\pm}}\bigg)\eta_{\pm}^{-(\gamma+N)}.\end{split}\end{equation}$$∙ ,Third, we conclude by Lemma 3.5, Lemma 3.6 (iii)−(iv) and a straightforward calculation that(4.33)limd(x)→0I3±(d(x))=0.$$\begin{equation}\lim_{d(x)\rightarrow0}I_{3\pm}(d(x))=0.\end{equation}$$Combining (4.31)-(4.33), we obtain (4.30) holds. By (b1), (b3) and (4.30), we see that there exists a sufficiently small positive constant δεδ1 such that for any x∈Ωδε(d(x))−(N+1)L˜N(d(x))(1+(B0−ε)(d(x))μ)≤b(x)≤(d(x))−(N+1)L˜N(d(x))(1+(B0+ε)(d(x))μ)$$(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\big(1+(B_{0}-\varepsilon)(d(x))^{\mu}\big)\leqb(x)\leq(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\big(1+(B_{0}+\varepsilon)(d(x))^{\mu}\big)$$and (4.4)-(4.5) hold here. Step 2. Letu_ε(x)=−η+ψ(r(d(x)))(1+(C+∗+ε)M(d(x))),x∈Ωδε,$$\underline{u}_{\varepsilon}(x)=-\eta_{+}\psi(r(d(x)))\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big),\,x\in\Omega_{\delta_{\varepsilon}},$$where r(d(x)) is given by (4.29). Theng(−u_ε(x))=g(η+ψ(r(d(x))))+η+ψ(r(d(x)))g′(Θ+(d(x)))(C+∗+ε)M(d(x)),x∈Ωδε,$$g(-\underline{u}_{\varepsilon}(x))=g(\eta_{+}\psi(r(d(x))))+\eta_{+}\psi(r(d(x)))g'(\Theta_{+}(d(x)))(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x)),\,x\in\Omega_{\delta_{\varepsilon}},$$where Θ+(d(x)) is given by (4.28). By Lemma 4.2, we have for any x∈Ωδε(4.34)det ( D 2 u _ ε ( x ) ) − b ( x ) g ( − u _ ε ( x ) ) ≥ − η + N ( N N + 1 ) N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) × L ~ N ( d ( x ) ) ( 1 + ( C + ∗ + ε ) ( N − 1 ) M ( d ( x ) ) + ν + ∗ ( d ( x ) ) ) [ ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) + β N N + 1 ( C + ∗ + ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( M ( d ( x ) ) ) β + 1 β + β N N + 1 ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( β + 1 ) N ( C + ∗ + ε ) N + 1 × ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 2 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( C + ∗ + ε ) × ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 1 β ( L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] × ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) ( 1 + ( B 0 + ε ) ( d ( x ) ) μ ) × [ g ( η + ψ ( r ( d ( x ) ) ) ) + η + ( C + ∗ + ε ) g ′ ( Θ + ( d ( x ) ) ) ψ ( r ( d ( x ) ) ) M ( d ( x ) ) ] = − η + N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) M ( d ( x ) ) { ( M ( d ( x ) ) ) − 1 × [ ( ( N N + 1 ) N ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ) m ^ + − g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ] + ( C + ∗ + ε ) [ ( N N + 1 ) N N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − η + ψ ( r ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) g ′ ( Θ + ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ] − ( B 0 + ε ) g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ ( M ( d ( x ) ) ) − 1 + ( M ( d ( x ) ) ) − 1 [ ( N N + 1 ) N ( N N + 1 − 1 N + 1 × ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] × m ^ + ∑ i = 1 N − 1 C N − 1 i ( − 1 ) i + 1 ( m + d ( x ) ) i ( 1 − d ( x ) m + ) N − 1 − ( B 0 + ε ) ( C + ∗ + ε ) η + ψ ( r ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) g ′ ( Θ + ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ + ( N N + 1 ) N ζ + ∗ ( d ( x ) ) ( M ( d ( x ) ) ) − 1 ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) } ≥ − η + N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) M ( d ( x ) ) ∑ i = 1 3 I i + ( d ( x ) ) > 0 , $$\begin{align} \nonumber&\mbox{det}(D^{2}\underline{u}_{\varepsilon}(x))-b(x)g(-\underline{u}_{\varepsilon}(x))\geq-\eta_{+}^{N}\bigg(\frac{N}{N+1}\bigg)^{N}(\psi'(r(d(x))))^{N-1}\psi''(r(d(x)))(d(x))^{-(N+1)}\\ \nonumber&\times\tilde{L}^{N}(d(x))\big(1+(C_{+}^{*}+\varepsilon)(N-1)\mathfrak{M}(d(x))+\nu_{+}^{*}(d(x))\big)\bigg[\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\\ \nonumber&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\\ \nonumber&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big)+\frac{\beta N}{N+1}(C_{+}^{*}+\varepsilon)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\ \nonumber&\times(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}+\frac{\beta N}{N+1}(C_{+}^{*}+\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\beta(\beta+1)N(C_{+}^{*}+\varepsilon)}{N+1}\\ \label{ffff} &\times\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}(\mathfrak{M}(d(x)))^{\frac{\beta+2}{\beta}}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\beta(C_{+}^{*}+\varepsilon)\\ \nonumber&\times\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\bigg(\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]\\ \nonumber&\times\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}-(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\big(1+(B_{0}+\varepsilon)(d(x))^{\mu}\big)\\ \nonumber&\times\big[g(\eta_{+}\psi(r(d(x))))+\eta_{+}(C_{+}^{*}+\varepsilon)g'(\Theta_{+}(d(x)))\psi(r(d(x)))\mathfrak{M}(d(x))\big]\\ \nonumber&=-\eta_{+}^{N}(\psi'(r(d(x))))^{N-1}\psi''(r(d(x)))(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\mathfrak{M}(d(x))\bigg\{(\mathfrak{M}(d(x)))^{-1}\\ \nonumber&\times\bigg[\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\ \nonumber&+\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg)\hat{m}_{+}-\frac{g(\eta_{+}\psi(r(d(x))))}{\eta_{+}^{N}g(\psi(r(d(x))))}\bigg]\\ \nonumber&+(C_{+}^{*}+\varepsilon)\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}N\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\\ \nonumber&-\frac{\eta_{+}\psi(r(d(x)))g'(\eta_{+}\psi(r(d(x))))}{g(\eta_{+}\psi(r(d(x))))}\frac{g'(\Theta_{+}(d(x)))}{g'(\eta_{+}\psi(r(d(x))))}\frac{g(\eta_{+}\psi(r(d(x))))}{\eta_{+}^{N}g(\psi(r(d(x))))}\bigg]\\ \nonumber&-(B_{0}+\varepsilon)\frac{g(\eta_{+}\psi(r(d(x))))}{\eta_{+}^{N}g(\psi(r(d(x))))}(d(x))^{\mu}(\mathfrak{M}(d(x)))^{-1}+(\mathfrak{M}(d(x)))^{-1}\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}\bigg(\frac{N}{N+1}-\frac{1}{N+1}\\ \nonumber&\times\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]\\ \nonumber&\times\frac{\hat{m}_{+}\sum_{i=1}^{N-1}C_{N-1}^{i}(-1)^{i+1}(m_{+}d(x))^{i}}{(1-d(x)m_{+})^{N-1}}\\ \nonumber&-(B_{0}+\varepsilon)(C_{+}^{*}+\varepsilon)\frac{\eta_{+}\psi(r(d(x)))g'(\eta_{+}\psi(r(d(x))))}{g(\eta_{+}\psi(r(d(x))))}\frac{g'(\Theta_{+}(d(x)))}{g'(\eta_{+}\psi(r(d(x))))}\frac{g(\eta_{+}\psi(r(d(x))))}{\eta_{+}^{N}g(\psi(r(d(x))))}(d(x))^{\mu}\\ \nonumber&+\bigg(\frac{N}{N+1}\bigg)^{N}\zeta_{+}^{*}(d(x))(\mathfrak{M}(d(x)))^{-1}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\bigg\}\\ \nonumber&\geq-\eta_{+}^{N}(\psi'(r(d(x))))^{N-1}\psi''(r(d(x)))(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\mathfrak{M}(d(x))\sum\limits_{i=1}^{3}I_{i+}(d(x))>0, \end{align}$$i.e. uε is a subsolution of Eq. (1.1) in Ωδε. Moreover, it follows by Lemma 4.3 that for i=1, ... ,NS i ( D 2 u _ ε ( x ) ) = η + i ( N N + 1 ) i ( ψ ′ ( r ( d ( x ) ) ) ) i ( d ( x ) ) − i L ~ i ( d ( x ) ) ( ∫ 0 d ( x ) L ~ ( s ) s d s ) − i N + 1 × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) + β ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β ) i S i ( ϵ 1 , ⋅ ⋅ ⋅ , ϵ N − 1 ) − η + i ( N N + 1 ) i ( ψ ′ ( r ( d ( x ) ) ) ) i − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( i + 1 ) L ~ i ( d ( x ) ) ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N − i N + 1 × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) + β ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β ) i − 1 [ ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) + β N N + 1 ( C + ∗ + ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( M ( d ( x ) ) ) β + 1 β + β N N + 1 ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( β + 1 ) N ( C + ∗ + ε ) N + 1 × ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 2 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( C + ∗ + ε ) × ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 1 β ( L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] S i − 1 ( ϵ 1 , ⋅ ⋅ ⋅ , ϵ N − 1 ) . $$\begin{split}&S_{i}(D^{2}\underline{u}_{\varepsilon}(x))=\eta_{+}^{i}\bigg(\frac{N}{N+1}\bigg)^{i}(\psi'(r(d(x))))^{i}(d(x))^{-i}\tilde{L}^{i}(d(x))\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{-\frac{i}{N+1}}\\&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))+\beta(C_{+}^{*}+\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\big)^{i}S_{i}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1})\\&-\eta_{+}^{i}\bigg(\frac{N}{N+1}\bigg)^{i}(\psi'(r(d(x))))^{i-1}\psi''(r(d(x)))(d(x))^{-(i+1)}\tilde{L}^{i}(d(x))\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N-i}{N+1}}\\&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))+\beta(C_{+}^{*}+\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\big)^{i-1}\bigg[\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\\&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\\&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big)+\frac{\betaN}{N+1}(C_{+}^{*}+\varepsilon)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\&\times(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}+\frac{\betaN}{N+1}(C_{+}^{*}+\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\beta(\beta+1)N(C_{+}^{*}+\varepsilon)}{N+1}\\&\times\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}(\mathfrak{M}(d(x)))^{\frac{\beta+2}{\beta}}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\beta(C_{+}^{*}+\varepsilon)\\&\times\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\bigg(\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]S_{i-1}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1}).\end{split}\quad\quad\quad\quad\quad\quad\quad\quad\quad$$This implies that we can adjust the above positive constant δε such that for any x∈ΩδεS i ( D 2 u _ ε ( x ) ) > 0 for i = 1 , ⋅ ⋅ ⋅ , N . $$S_{i}(D^{2}\underline{u}_{\varepsilon}(x))>0 \,\mbox{for}\,i=1,\cdot\cdot\cdot,N.$$We obtain by Lemma 4.4 that D2uε is positive definite in Ωδε.Letu‾ε(x)=−η−ψ(r(d(x)))(1+(C−∗−ε)M(d(x))),x∈Ωδε$$\overline{u}_{\varepsilon}(x)=-\eta_{-}\psi(r(d(x)))\big(1+(C_{-}^{*}-\varepsilon)\mathfrak{M}(d(x))\big),\,x\in\Omega_{\delta_{\varepsilon}}$$By the same calculation as (4.34), we obtaindet ( D 2 u ¯ ε ( x ) ) − b ( x ) g ( − u ¯ ε ( x ) ) ≤ − η − N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) M ( d ( x ) ) ∑ i = 1 3 I i − ( d ( x ) ) < 0 , $$\begin{split}&\mbox{det}(D^{2}\overline{u}_{\varepsilon}(x))-b(x)g(-\overline{u}_{\varepsilon}(x))\\&\leq-\eta_{-}^{N}(\psi'(r(d(x))))^{N-1}\psi''(r(d(x)))(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\mathfrak{M}(d(x))\sum\limits_{i=1}^{3}I_{i-}(d(x))<0,\end{split}$$i.e. uε is a supersolution of Eq. (1.1) in Ωδε.Let u be the unique strictly convex solution to problem (1.1). Through the same argument as Theorem 1.1, we see that there exists a large constant M > 0 such thatu_ε(x)−Md(x)≤u(x)≤u‾ε(x)+Md(x),x∈Ωδε.$$\underline{u}_{\varepsilon}(x)-Md(x)\leq u(x)\leq\overline{u}_{\varepsilon}(x)+Md(x),\,x\in\Omega_{\delta_{\varepsilon}}.$$Hence, we have for any x∈ ΩδεC+∗+ε+M(M(d(x)))−1d(x)η+ψ(r(d(x)))≥(−u(x)η+ψ(r(d(x)))−1)(M(d(x)))−1;C−∗−ε−M(M(d(x)))−1d(x)η−ψ(r(d(x)))≤(−u(x)η−ψ(r(d(x)))−1)(M(d(x)))−1.$$\begin{split}C_{+}^{*}+\varepsilon+\frac{M(\mathfrak{M}(d(x)))^{-1}d(x)}{\eta_{+}\psi(r(d(x)))}&\geq\bigg(\frac{-u(x)}{\eta_{+}\psi(r(d(x)))}-1\bigg)(\mathfrak{M}(d(x)))^{-1};\\C_{-}^{*}-\varepsilon-\frac{M(\mathfrak{M}(d(x)))^{-1}d(x)}{\eta_{-}\psi(r(d(x)))}&\leq\bigg(\frac{-u(x)}{\eta_{-}\psi(r(d(x)))}-1\bigg)(\mathfrak{M}(d(x)))^{-1}.\end{split}$$We conclude from (3.17)-(3.18), Proposition 2.5 (i)−(ii) thatC+∗+ε≥lim supd(x)→0(−u(x)η+ψ(r(d(x)))−1)(M(d(x)))−1;C−∗−ε≤lim infd(x)→0(−u(x)η−ψ(r(d(x)))−1)(M(d(x)))−1.$$\begin{split}C_{+}^{*}+\varepsilon&\geq\limsup_{d(x)\rightarrow0}\bigg(\frac{-u(x)}{\eta_{+}\psi(r(d(x)))}-1\bigg)(\mathfrak{M}(d(x)))^{-1};\\C_{-}^{*}-\varepsilon&\leq\liminf_{d(x)\rightarrow0}\bigg(\frac{-u(x)}{\eta_{-}\psi(r(d(x)))}-1\bigg)(\mathfrak{M}(d(x)))^{-1}.\end{split}$$Letting ε → 0 , the proof is finished. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Nonlinear Analysis de Gruyter http://www.deepdyve.com/lp/de-gruyter/refined-second-boundary-behavior-of-the-unique-strictly-convex-IrCCRGdYzB

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Publisher: de Gruyter
Copyright: © 2021 HaitaoWan et al., published by De Gruyter
ISSN: 2191-9496
eISSN: 2191-950X
DOI: 10.1515/anona-2022-0199
Publisher site: See Article on Publisher Site

Abstract

1Introduction and main resultsThis presentation is to establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge–Ampère equation(1.1) det(D2u)=b(x)g(−u),u<0 in Ω and u=0 on ∂Ω,$$\begin{equation}\mbox{ det}(D^{2} u) =b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and }u=0 \mbox{ on }\partial\Omega,\end{equation}$$where Ω is a bounded, smooth and strictly convex domain in ℝN (N ≥ 2), andD2u(x)=(∂2u(x)∂xi∂xj)N×N$$D^{2}u(x)=\bigg(\frac{\partial^{2} u(x)}{\partial x_{i}\partialx_{j}}\bigg)_{N\times N}$$denotes the Hessian of u and D2u is the so called Monge–Ampère operator. The nonlinearity g satisfies(g1) g ∈ C1((0, ∞), (0, ∞)) is decreasing on (0, ∞) and lim t → 0 + g ( t ) = ∞ $ \lim\limits_{t\rightarrow0^{+}}g(t)=\infty $;(g2) there exist a constant γ>1 and some function f ∈ C1(0, a1)∩ C[0, a1) for a sufficiently small constant a1 > 0 such that−tg′(t)g(t):=γ+f(t) withlimt→0+f(t)=0,$$\frac{-tg'(t)}{g(t)}:=\gamma+f(t) \mbox{ with}\lim_{t\rightarrow0^{+}}f(t)=0,$$i.e.,g(t)=c0t−γexp∫ta1f(s)sds,c0=g(a1)a1γ,$$g(t)=c_{0}t^{-\gamma}\exp\left(\int_{t}^{a_{1}}\frac{f(s)}{s}ds\right),\,c_{0}=g(a_{1})a_{1}^{\gamma},$$where the function f satisfies(S1) f ≡ 0 on (0, a1] (or (S2) f(t)≠0, ∀ t ∈ (0, a] for some a ≤ a1).If (S2) holds in (g2) , then we suppose(g3) there exists θ ≥ 0 such thatlim t → 0 + t f ′ ( t ) f ( t ) = θ ≥ 0. $$\lim\limits_{t\rightarrow0^{+}}\frac{tf'(t)}{f(t)}=\theta\geq0.$$If θ=0 in (g3), then we further suppose(g4) there exist β ∈ ℝ+ and σ∈R$ \sigma\in \mathbb{R} $such thatlimt→0+(−lnt)βf(t)=σ.$$\lim_{t\rightarrow0^{+}}(-\ln t)^{\beta}f(t)=\sigma.$$The weight b satisfies(b1) b ∈ C∞(Ω) is positive in Ωand one of the following two conditions(b2) there exist k ∈ Λ, B0∈R$ B_{0}\in\mathbb{R} $and μ ∈ ℝ+ such thatb(x)=kN+1(d(x))(1+B0(d(x))μ+o((d(x))μ)),d(x)→0,$$b(x)=k^{N+1}(d(x))\big(1+B_{0}(d(x))^{\mu}+o((d(x))^{\mu})\big),\,d(x)\rightarrow0,$$where d(x)≔ dist(x, ∂Ω), x ∈ Ω, Λ denotes the set of all of positive monotonic functions in C1(0, δ0) ∩ L (0, δ0) which satisfylimt→0+ddt(K(t)k(t))=Dk≥0,K(t)=∫0tk(s)ds$$\lim_{t\rightarrow0^{+}}\frac{d}{dt}\bigg(\frac{K(t)}{k(t)}\bigg)=D_{k}\geq0,\,\,\,K(t)=\int_{0}^{t}k(s)ds$$and(b3) there exist L˜∈L$ \tilde{L}\in \mathcal{L} $, B0∈R$ B_{0}\in \mathbb{R} $and μ ∈ ℝ+ such thatb(x)=(d(x))−(N+1)L˜N(d(x))(1+B0(d(x))μ+o((d(x))μ)),d(x)→0,$$b(x)=(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\big(1+B_{0}(d(x))^{\mu}+o((d(x))^{\mu})\big),\,d(x)\rightarrow0,$$where L$ \mathcal {L} $denotes the set of all of positive functionsdefined on (0, t0] byL ~ ( t ) := c exp ⁡ ( ∫ t t 0 y ( s ) s d s ) , t ∈ ( 0 , t 0 ] , $$\tilde{L}(t):=c\exp\bigg(\int\limits_{t}^{t_{0}}\frac{y(s)}{s}ds\bigg),\,t\in(0,t_{0}],$$where c ∈ ℝ+, y ∈ C (0, t0] andlimt→0+y(t)=0$ \lim_{t\rightarrow0^{+}}y(t)=0 $.The set Λ in (b2) was first introduced by Cîrstea and Rădulescu [6]- [8] for non-decreasing functions and by Mohammed [33] for non-increasing functions to study the exact boundary behavior and uniqueness of boundary blow-up elliptic problems. When b satisfies (b2) with Dk > 0, we see by Lemma 3.1 (iii) -(iv) that b may be singular on the boundary with the index(1−Dk)(N+1)Dk>−(N+1).$$\frac{(1-D_{k})(N+1)}{D_{k}}>-(N+1).$$The condition (b3) implies that b is critical singular with the index −(N + 1).problem (1.1) has a wide range of applications in Riemannian geometry and optical physics and one important geometric application is to structure a Riemannian metric in Ω that is invariant under projective transformations. When g(t)=t−(N+2), t > 0 and b ≡ 1 in Ω, Nirenberg [38], Loewner and Nirenberg [31] for N = 2, Cheng and Yau [5] for N ≥ 2 studied the existence and uniqueness of solutions to problem (1.1). In particular, Cheng and Yau [5] showed that if Ω is convex and bounded but not necessarily strictly convex then problem (1.1) possesses a unique solution u∈C∞(Ω)∩C(Ωˉ)$ u\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $which is negative in Ω. When g(t)=t−γ (t > 0) with γ>1 and b∈C∞(Ωˉ)$ b\in C^{\infty}(\bar{\Omega}) $with b(x) > 0 for all x ∈ Ω, Lazer and McKenna [27] proved the existence and uniqueness of solutions to problem (1.1). Moreover, they also obtained the following global estimatec1(d(x))N+1N+γ≤u(x)≤c2(d(x))N+1N+γ,x∈Ω.$$c_{1}(d(x))^{\frac{N+1}{N+\gamma}}\leq u(x)\leqc_{2}(d(x))^{\frac{N+1}{N+\gamma}},\,x\in\Omega.$$When b satisfies (b1) and g:(0, ∞) → (0, ∞) is a non-increasing, smooth function, Mohammed [34] showed that problem (1.1) has a strictly convex solution u∈C∞(Ω)∩C(Ωˉ)$ u\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $if and only if the problem(1.2) det(D2u)=b(x) in Ω and u=0 on ∂Ω$$\begin{equation}\mbox{ det}(D^{2}u)=b(x) \mbox{ in }\Omega \mbox{ and }u=0 \mbox{ on }\partial\Omega\end{equation}$$has a strictly convex solution v∈C∞(Ω)∩C(Ωˉ)$ v\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $, where b may be singular or may vanish on ∂Ω. In particular, the author showed that(i1) if b∈C∞(Ωˉ)$ b\in C^{\infty}(\bar{\Omega}) $is positive in Ωˉ$ \bar{\Omega} $, then problem (1.1) has a strictly convex solution u∈C∞(Ω)∩C(Ωˉ)$ u\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $;(i2) if limt→0+g(t)=∞$ \lim_{t\rightarrow0^{+}}g(t)=\infty $,then problem (1.1) has a unique strictly convex solution u∈C∞(Ω)∩C(Ωˉ)$ u\in C^{\infty}(\Omega)\cap C(\bar{\Omega}) $and the solution usatisfiesc1ϕ(d(x))≤−u(x)≤c2ϕ(d(x)) and |∇u(x)|≤c2ϕ(d(x))d(x) near ∂Ω,$$c_{1}\phi(d(x))\leq-u(x)\leq c_{2}\phi(d(x)) \mbox{ and }|\nablau(x)|\leq c_{2}\frac{\phi(d(x))}{d(x)} \mbox{ near }\partial\Omega,$$where c1, c2 are positive constants and ϕ is uniquely determined by∫ 0 ϕ ( t ) ( G ( s ) ) − ( N + 1 ) d s = t , G ( t ) = ∫ t t ^ g ( s ) d s , t ∈ ( 0 , t ^ ) , t ^ ∈ ( 0 , ∞ ] ; $$\int\limits_{0}^{\phi(t)}(G(s))^{-(N+1)}ds=t,\,G(t)=\int\limits_{t}^{\hat{t}}g(s)ds,\,t\in(0,\hat{t}),\,\hat{t}\in (0, \infty];$$(i3) if b(x) ⩽ C(d (x))δ−N−1 for somepositive constants δ and C, then problem (1.2) has a strictly convex solution;(i4) if b(x)=C(d (x))−(N+1) for somepositive constant C, then problem (1.2) has no strictlyconvex solution.Later, Yang and Chang [47] extended the above results (i3) -(i4) to the following cases:(i5) if b(x) ⩽ C(d (x))−(N+1)(−ln d(x))−qnear ∂Ω for some q > N and C > 0, then problem (1.2) has a strictly convex solution;(i6) if b(x)=C(d (x))−(N+1)(−lnd(x))−N near ∂Ω for some C > 0, then problem (1.2) has no strictly convex solution.Let P∈C1(0,∞)$ \mathscr{P}\in C^{1}(0, \infty) $satisfy P′(t)<0$ \mathscr{P}'(t)<0 $and limt→0+P(t)=∞$ \lim_{t\rightarrow0^{+}}\mathscr{P}(t)=\infty $and define P(t)=∫t1P(s)ds$ \mathfrak{P}(t)=\int_{t}^{1}\mathscr{P}(s)ds $. Recently, under the hypothesis of (b1), Zhang and Du [49] obtain the following results:(i7) if b(x)≤P(d(x))$ b(x)\leq \mathscr{P}(d(x)) $near∂Ω and∫01(P(s))1/Nds<∞$ \int_{0}^{1}(\mathfrak{P}(s))^{1/N}ds<\infty $, then problem (1.2) has a strictly convex solution;(i8) if b(x)≥P(d(x))$ b(x)\geq \mathscr{P}(d(x)) $and∫01(P(s))1/Nds=∞$ \int_{0}^{1}(\mathfrak{P}(s))^{1/N}ds=\infty $, then problem (1.2) has no strictly convex solution.The above facts imply that problem (1.2) has a strictly convex solution if b satisfies (b1) andb(x)≤CkN+1(d(x)) near ∂Ω orb(x)≤C(d(x))−(N+1)L˜N(d(x)) near ∂Ω,$$b(x)\leq Ck^{N+1}(d(x)) \mbox{ near }\partial\Omega \mbox{ or}b(x)\leq C (d(x))^{-(N+1)}\tilde{L}^{N}(d(x)) \mbox{ near }\partial\Omega,$$where C is a positive constant, k ∈ Λ in (b2) and L˜∈L$ \tilde{L}\in\mathcal {L} $in (b3) with(1.3)∫ 0 t L ~ ( s ) s d s < ∞ . $$\begin{equation}\int\limits_{0}^{t}\frac{\tilde{L}(s)}{s}ds<\infty.\end{equation}$$In [28], Li and Ma studied the existence and the firstboundary behavior of the strictly convex solutions to problem (1.1) by using regularity theory and sub-supersolution method.In particular, when b ∈ C3(Ω) is positive in Ω and satisfies(b01) there exist k ∈ Λ and positiveconstants b1 and b2 such thatb1:=lim infd(x)→0b(x)kN+1(d(x))≤lim supd(x)→0b(x)kN+1(d(x))=:b2,$$b_{1}:=\liminf_{d(x)\rightarrow0}\frac{b(x)}{k^{N+1}(d(x))}\leq\limsup_{d(x)\rightarrow0}\frac{b(x)}{k^{N+1}(d(x))}=:b_{2},$$g satisfies (g1) and(g01)lim t → 0 + ( G ( t ) ) 1 / ( N + 1 ) ∫ 0 t ( G ( s ) ) − 1 / ( N + 1 ) d s = D g , G ( t ) = ∫ t a 1 g ( s ) d s for some a 1 > 0 , $$\lim\limits_{t\rightarrow0^{+}}(G(t))^{1/(N+1)}\int\limits_{0}^{t}(G(s))^{-1/(N+1)}ds=D_{g},\,\,\,G(t)=\int\limits_{t}^{a_{1}}g(s)ds\mbox{ for some}a_{1}>0,$$they showed that the unique strictly convex solution u to problem (1.1) satisfies1≤lim infd(x)→0−u(x)ψ(ϑ1K(d(x)))≤lim supd(x)→0−u(x)ψ(ϑ2K(d(x)))≤1,$$1\leq\liminf_{d(x)\rightarrow0}\frac{-u(x)}{\psi(\vartheta_{1}K(d(x)))}\leq\limsup_{d(x)\rightarrow0}\frac{-u(x)}{\psi(\vartheta_{2}K(d(x)))}\leq1,$$where ψ is uniquely determined by(1.4)∫ 0 ψ ( t ) ( ( N + 1 ) G ( s ) ) − 1 / ( N + 1 ) d s = t , $$\begin{equation}\int\limits_{0}^{\psi(t)}((N+1)G(s))^{-1/(N+1)}ds=t,\end{equation}$$ϑ1=(b1mˆ−(1−Dg−1(1−Dk)))1/(N+1) and ϑ2=(b2mˆ+(1−Dg−1(1−Dk)))1/(N+1)$$\vartheta_{1}=\bigg(\frac{b_{1}}{\hat{m}_{-}(1-D_{g}^{-1}(1-D_{k}))}\bigg)^{1/(N+1)}\mbox{ and }\vartheta_{2}=\bigg(\frac{b_{2}}{\hat{m}_{+}(1-D_{g}^{-1}(1-D_{k}))}\bigg)^{1/(N+1)}$$with(1.5)mˆ−:=maxxˉ∈∂ΩωN−1(xˉ) and mˆ+:=minxˉ∈∂ΩωN−1(xˉ),$$\begin{equation}\hat{m}_{-}:=\max_{\bar{x}\in\partial\Omega}\omega_{N-1}(\bar{x}) \mbox{ and }\hat{m}_{+}:=\min_{\bar{x}\in\partial\Omega}\omega_{N-1}(\bar{x}),\end{equation}$$whereωN−1(xˉ)=∏i=1N−1κi(xˉ)$$\omega_{N-1}(\bar{x})=\prod_{i=1}^{N-1}\kappa_{i}(\bar{x})$$denotes the (N − 1)-th curvature at xˉ$ \bar{x} $and κ1(xˉ),⋅⋅⋅,κN−1(xˉ)$ \kappa_{1}(\bar{x}),\cdot\cdot\cdot,\kappa_{N-1}(\bar{x}) $denote the principal curvatures of ∂Ω at xˉ$ \bar{x} $. In [51], Zhang showed that if b satisfies (b1) and (b01), g satisfies (g1) and(g02)lim t → 0 + ( ( g ( t ) ) 1 / N ) ′ ∫ 0 t ( g ( s ) ) − 1 / N d s = − C g , $$\lim\limits_{t\rightarrow0^{+}}((g(t))^{1/N})'\int\limits_{0}^{t}(g(s))^{-1/N}ds=-C_{g},$$and NDk + (1 + N)Cg > 1+N, then the unique strictly convex solution u to problem (1.1) satisfiesϑ31−Cg:=lim infd(x)→0−u(x)ϕg((K(d(x)))(N+1)/N)≤lim supd(x)→0−u(x)ϕg((K(d(x)))(N+1)/N)=:ϑ41−Cg,$$\vartheta_{3}^{1-C_{g}}:=\liminf_{d(x)\rightarrow0}\frac{-u(x)}{\phi_{g}((K(d(x)))^{(N+1)/N})}\leq\limsup_{d(x)\rightarrow0}\frac{-u(x)}{\phi_{g}((K(d(x)))^{(N+1)/N})}=:\vartheta_{4}^{1-C_{g}},$$where ϕg is uniquely determined by∫ 0 ϕ g ( t ) ( N g ( s ) ) − 1 / N d s = t , t > 0 , $$\int\limits_{0}^{\phi_{g}(t)}(Ng(s))^{-1/N}ds=t,\,t>0,$$ϑ3=((NN+1)Nb1mˆ−((1+N)Cg+NDk−1−N))1/N$$\vartheta_{3}=\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{b_{1}}{\hat{m}_{-}\big((1+N)C_{g}+ND_{k}-1-N\big)}\bigg)^{1/N}$$andϑ4=((NN+1)Nb2mˆ+((1+N)Cg+NDk−1−N))1/N.$$\vartheta_{4}=\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{b_{2}}{\hat{m}_{+}\big((1+N)C_{g}+ND_{k}-1-N\big)}\bigg)^{1/N}.$$Especially, if (b01) is replaced by the following condition(b02) there exist L˜∈L$ \tilde{L}\in\mathcal {L} $with (1.3) and positive constants b1 and b2 such thatb1:=lim infd(x)→0b(x)(d(x))−(N+1)L˜N(d(x))≤lim supd(x)→0b(x)(d(x))−(N+1)L˜N(d(x))=:b2,$$b_{1}:=\liminf_{d(x)\rightarrow0}\frac{b(x)}{(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))}\leq\limsup_{d(x)\rightarrow0}\frac{b(x)}{(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))}=:b_{2},$$Zhang [51] showed that the unique strictly convex solution u to problem (1.1) satisfiesϑ51−Cg≤lim infd(x)→0−u(x)ϕg(∫0d(x)L˜(s)sds)≤lim supd(x)→0−u(x)ϕg(∫0d(x)L˜(s)sds)≤ϑ61−Cg,$$\vartheta_{5}^{1-C_{g}}\leq\liminf_{d(x)\rightarrow0}\frac{-u(x)}{\phi_{g}\big(\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\big)}\leq\limsup_{d(x)\rightarrow0}\frac{-u(x)}{\phi_{g}\big(\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\big)}\leq\vartheta_{6}^{1-C_{g}},$$whereϑ5=(b1mˆ−N)1/N and ϑ6=(b2mˆ+N)1/N.$$\vartheta_{5}=\bigg(\frac{b_{1}}{\hat{m}_{-}N}\bigg)^{1/N} \mbox{ and }\vartheta_{6}=\bigg(\frac{b_{2}}{\hat{m}_{+}N}\bigg)^{1/N}.$$Then, Sun and Feng [43] and Li and Ma [29] generalized the above boundary behavior results to the case of the following Hessian equation for i=1, ···, NSi(D2u)=b(x)g(−u),u<0 in Ω and u=0 on ∂Ω,$$S_{i}(D^{2} u) =b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and }u=0\mbox{ on }\partial\Omega,$$whereS i ( D 2 u ) = S i ( λ 1 , ⋅ ⋅ ⋅ , λ N ) = ∑ 1 ≤ j 1 < ⋅ ⋅ ⋅ < j i ≤ N λ j 1 ⋅ ⋅ ⋅ λ j i $$S_{i}(D^{2}u)=S_{i}(\lambda_{1},\cdot\cdot\cdot,\lambda_{N})=\sum_{1\leqj_{1}<\cdot\cdot\cdot<j_{i}\leqN}\lambda_{j_{1}}\cdot\cdot\cdot\lambda_{j_{i}}$$and λ1, ···, λN are the eigenvalues of D2 u. Furthermore S0(λ) ≡ 1 for λ ∈ ℝN. Espically, Li and Ma [29] also studied the existence and uniqueness of viscosity solution to the problem. For related insights on the existence, regularity and asymptotic behavior of solutions to the Monge-Ampère equations, please refer to [4], [11], [19], [21]- [25], [30], [35]- [37], [44]- [45] and the references therein. When the Monge-Ampère operator (det(D2u)) is replaced by the Laplace operator (Δ), many papers have been dedicated to resolving existence, uniqueness and asymptotic behavior issues for solutions, please refer to [1]- [2], [10], [12]- [17], [26], [39], [46], [48], [50] and the references therein.In this paper, by making a complete and detailed analysis to some indexes in various cases, we establish the exact second boundary behavior of the unique strictly convex solution to problem (1.1), which is quite different from the first behavior of this solution. For all we know, in literature there aren't articles on the second boundary behavior of the strictly convex solution to problem (1.1).To our aims, we define the following subclasses of Λ and L$ \mathcal {L} $as follows:Λ 1 := { k ∈ Λ : lim t → 0 + t − 1 [ d d t ( K ( t ) k ( t ) ) − D k ] = E 1 , k } ; Λ 2 , β := { k ∈ Λ : lim t → 0 + ( − ln ⁡ t ) β [ d d t ( K ( t ) k ( t ) ) − D k ] = E 2 , k } ; L β := { L ~ ∈ L : lim t → 0 + ( − ln ⁡ t ) β y ( t ) = E 3 } , $$\begin{split}&\Lambda_{1}:=\bigg\{k\in\Lambda:\lim\limits_{t\rightarrow0^{+}}t^{-1}\bigg[\frac{d}{dt}\bigg(\frac{K(t)}{k(t)}\bigg)-D_{k}\bigg]=E_{1,k}\bigg\};\\&\Lambda_{2,\beta}:=\bigg\{k\in\Lambda:\lim\limits_{t\rightarrow0^{+}}(-\ln t)^{\beta}\bigg[\frac{d}{dt}\bigg(\frac{K(t)}{k(t)}\bigg)-D_{k}\bigg]=E_{2,k}\bigg\};\\&\mathfrak{L}_{\beta}:=\{\tilde{L}\in\mathcal {L}:\lim\limits_{t\rightarrow0^{+}}(-\ln t)^{\beta}y(t)=E_{3}\},\end{split}$$where β is a positive constant and the relation between L˜$ \tilde{L} $and y is given in (b3). Our results are summarized as follows and mˆ±$ \hat{m}_{\pm} $(given in Theorems 1.1-1.3) are defined by (1.5).Theorem 1.1Let b satisfy (b1) -(b2) with (γ+N)Dk>N + 1, g satisfy (g1) -(g2).(I) When (S1) holds (or (S2) and (g3) -(g4) hold with θ=0 in (g3)), we have(i) If k ∈ Λ1, and then the unique strictly convex solution u to problem (1.1) satisfies(1.6)ξ−ψ(K(d(x)))(1+C−(−lnd(x))−β+o((−lnd(x))−β))≤−u(x)≤ξ+ψ(K(d(x)))(1+C+(−lnd(x))−β+o((−lnd(x))−β)),d(x)→0,$$\begin{equation}\begin{split}&\xi_{-}\psi(K(d(x)))\big(1+C_{-}(-\ln d(x))^{-\beta}+o((-\lnd(x))^{-\beta})\big)\\&\leq -u(x) \leq \xi_{+}\psi(K(d(x)))\big(1+C_{+}(-\lnd(x))^{-\beta}+o((-\ln d(x))^{-\beta})\big),\,d(x)\rightarrow0,\end{split}\end{equation}$$where ψ is uniquely determined by (1.4) and(1.7)ξ±=(((γ+N)Dk−(N+1))mˆ±γ−1)−1/(γ+N),$$\begin{equation}\xi_{\pm}=\bigg(\frac{\big((\gamma+N)D_{k}-(N+1)\big)\hat{m}_{\pm}}{\gamma-1}\bigg)^{-1/(\gamma+N)},\end{equation}$$C ± = − ( ( γ + N ) D k N + 1 ) β C ± , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , 0 , i f ( S 1 ) h o l d s , $$C_{\pm}=\begin{cases}-\big(\frac{(\gamma+N)D_{k}}{N+1}\big)^{\beta}\mathfrak{C}_{\pm},\, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})}\,{ hold\, with}\,\theta=0,\\0,\,&\,{ if }\,\mathbf{(S_{1})} \,{ holds}\,,\end{cases}$$whereC ± = A ± γ + N , i f A + ≤ 0 a n d A − ≤ 0 , C + = A + γ + N m ^ − m ^ + a n d C − = A − γ + N m ^ + m ^ − , i f A + ≥ 0 a n d A − ≥ 0 , C + = A + γ + N m ^ − m ^ + a n d C − = A − γ + N , i f A + > 0 a n d A − < 0 , C + = A + γ + N a n d C − = A − γ + N m ^ + m ^ − , i f A + < 0 a n d A − > 0 $$\begin{align}\begin{cases}\mathfrak{C}_{\pm}=\frac{A_{\pm}}{\gamma+N},\, &\,{ if }\,A_{+}\leq0 \,{ and }\,A_{-}\leq0,\\\mathfrak{C}_{+}=\frac{A_{+}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\mathfrak{C}_{-}=\frac{A_{-}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,A_{+}\geq0 \,{ and }\,A_{-}\geq0,\\\mathfrak{C}_{+}=\frac{A_{+}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\mathfrak{C}_{-}=\frac{A_{-}}{\gamma+N},\,&\,{ if }\,A_{+}>0 \,{ and }\,A_{-}<0,\\\mathfrak{C}_{+}=\frac{A_{+}}{\gamma+N} \,{ and }\,\mathfrak{C}_{-}=\frac{A_{-}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,A_{+}<0 \,{ and }\,A_{-}>0\end{cases}\end{align}$$with(1.8)A±=(C0+1γ+Nlnmˆ±−1)σ,$$\begin{equation}A_{\pm}=\bigg(C_{0}+\frac{1}{\gamma+N}\ln\hat{m}_{\pm}^{-1}\bigg)\sigma,\end{equation}$$and(1.9)C0=(N+1)(1−Dk)((γ+N)Dk−(N+1))(γ−1)−(γ+N)−1ln(γ+N)Dk−(N+1)γ−1.$$\begin{equation}C_{0}=\frac{(N+1)(1-D_{k})}{\big((\gamma+N)D_{k}-(N+1)\big)(\gamma-1)}-(\gamma+N)^{-1}\ln\frac{(\gamma+N)D_{k}-(N+1)}{\gamma-1}.\end{equation}$$(ii) If k ∈ Λ2,β (β is the same as the one in(g4)), then (1.6) still holds, whereC ± = − C ^ ± , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , − D ^ ± , i f ( S 1 ) h o l d s , $$\begin{align}C_{\pm}=\begin{cases}-\hat{C}_{\pm},\, &\,{ if }\, \mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})}\,{ hold \,with}\,\theta=0,\\-\hat{D}_{\pm},\,&\,{ if }\,\mathbf{(S_{1})} \,{ holds},\end{cases}\end{align}$$andC ^ ± = A ± + B γ + N , i f A + + B ≤ 0 a n d A − + B ≤ 0 , C ^ + = A + + B γ + N m ^ − m ^ + a n d C ^ − = A − + B γ + N m ^ + m ^ − , i f A + + B ≥ 0 a n d A − + B ≥ 0 , C ^ + = A + + B γ + N m ^ − m ^ + a n d C ^ − = A − + B γ + N , i f A + + B > 0 a n d A − + B < 0 , C ^ + = A + + B γ + N a n d C ^ − = A − + B γ + N m ^ + m ^ − , i f A + + B < 0 a n d A − + B > 0 , $$\begin{align}\begin{cases}\hat{C}_{\pm}=\frac{\mathfrak{A}_{\pm}+\mathfrak{B}}{\gamma+N},\, &\,{ if }\,\mathfrak{A}_{+}+\mathfrak{B}\leq0 \,{ and }\,\mathfrak{A}_{-}+\mathfrak{B}\leq0,\\\hat{C}_{+}=\frac{\mathfrak{A}_{+}+\mathfrak{B}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\hat{C}_{-}=\frac{\mathfrak{A}_{-}+\mathfrak{B}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,\mathfrak{A}_{+}+\mathfrak{B}\geq0 \,{ and }\,\mathfrak{A}_{-}+\mathfrak{B}\geq0,\\\hat{C}_{+}=\frac{\mathfrak{A}_{+}+\mathfrak{B}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\hat{C}_{-}=\frac{\mathfrak{A}_{-}+\mathfrak{B}}{\gamma+N},\,&\,{ if }\,\mathfrak{A}_{+}+\mathfrak{B}>0 \,{ and }\,\mathfrak{A}_{-}+\mathfrak{B}<0,\\\hat{C}_{+}=\frac{\mathfrak{A}_{+}+\mathfrak{B}}{\gamma+N} \,{ and }\,\hat{C}_{-}=\frac{\mathfrak{A}_{-}+\mathfrak{B}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,\mathfrak{A}_{+}+\mathfrak{B}<0 \,{ and }\,\mathfrak{A}_{-}+\mathfrak{B}>0,\end{cases}\end{align}$$(1.10)D ^ + = B γ + N m ^ − m ^ + a n d D ^ − = B γ + N m ^ + m ^ − , i f B ≥ 0 , D ^ + = D ^ − = B γ + N , i f B < 0 $$\begin{equation}\begin{cases}\hat{D}_{+}=\frac{\mathfrak{B}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}} \,{ and }\,\hat{D}_{-}=\frac{\mathfrak{B}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}} ,\, &\,{ if }\,\mathfrak{B}\geq0,\\\hat{D}_{+}=\hat{D}_{-}=\frac{\mathfrak{B}}{\gamma+N},\,& \,{ if }\,\mathfrak{B}<0\end{cases}\end{equation}$$with(1.11)A ± = ( ( γ + N ) D k N + 1 ) β A ± a n d B = ( γ + N ) E 2 , k ( γ + N ) D k − ( N + 1 ) , $$\begin{equation}\mathfrak{A}_{\pm}=\bigg(\frac{(\gamma+N)D_{k}}{N+1}\bigg)^{\beta}A_{\pm}\,{ and }\,\mathfrak{B}=\frac{(\gamma+N)E_{2,k}}{(\gamma+N)D_{k}-(N+1)},\end{equation}$$where A ± are given by (1.8).(II) When (S2) and (g3) hold with θ > 0 in (g3) and k ∈ Λ2,β, then the unique strictly convexsolution u to problem (1.1) satisfies (1.6) with C±=−Dˆ±,$ C_{\pm}=-\hat{D}_{\pm}, $where Dˆ±$ \hat{D}_{\pm} $are given by(1.10).Corollary 1.1In Theorem 1.1, if Ω is a ball with radius R and center x0, then(I) When (S1) holds (or(S2) and (g3) -(g4) hold with θ = 0 in (g3)), we have(i) If k ∈ Λ1, then the unique strictly convex solution u to problem (1.1) satisfies(1.12)−u(x)=ξR1ψ(K(R−r))(1+CR1(−ln(R−r))β+o((−ln(R−r))β)),r→R,$$\begin{equation}-u(x)=\xi_{R}^{1}\psi(K(R-r))\big(1+C_{R}^{1}(-\ln(R-r))^{\beta}+o((-\ln (R-r))^{\beta})\big),\,r\rightarrow R,\end{equation}$$where r=∣x − x0∣,(1.13)ξR1=((γ+N)Dk−(N+1)(γ−1)RN−1)−1/(γ+N)$$\begin{equation}\xi_{R}^{1}=\bigg(\frac{(\gamma+N)D_{k}-(N+1)}{(\gamma-1)R^{N-1}}\bigg)^{-1/(\gamma+N)}\end{equation}$$andC R 1 = C ^ R 1 , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , 0 , i f ( S 1 ) h o l d s , $$C_{R}^{1}=\begin{cases}\hat{C}_{R}^{1},\, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})}\,{ hold\, with}\,\theta=0,\\0,\,&\,{ if }\, \mathbf{(S_{1})} \,{ holds},\,\end{cases}$$where(1.14)CˆR1=−((γ+N)DkN+1)β(C0+N−1γ+NlnR)σγ+N$$\begin{equation}\hat{C}_{R}^{1}=-\bigg(\frac{(\gamma+N)D_{k}}{N+1}\bigg)^{\beta}\frac{\big(C_{0}+\frac{N-1}{\gamma+N}\lnR\big)\sigma}{\gamma+N}\end{equation}$$and C0 is given by (1.9).(ii) If k ∈ Λ2,β (β is the same as the one in (g4)), then (1.12) still holds, whereC R 1 = C ^ R 1 − E 2 , k ( γ + N ) D k − ( N + 1 ) , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , − E 2 , k ( γ + N ) D k − ( N + 1 ) , i f ( S 1 ) h o l d s , $$C_{R}^{1}=\begin{cases}\hat{C}_{R}^{1}-\frac{E_{2,k}}{(\gamma+N)D_{k}-(N+1)},\, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})}\,{ hold\, with}\,\theta=0,\\-\frac{E_{2,k}}{(\gamma+N)D_{k}-(N+1)},\,&\,{ if }\,\mathbf{(S_{1})} \,{ holds}\,,\end{cases}$$where CˆR1$ \hat{C}_{R}^{1} $is given by (1.14) and C0 isgiven by (1.9).(II) When (S2) and (g3) hold with θ > 0 in (g3) and k ∈ Λ2,β, then the unique strictly convex solution u to problem (1.1) satisfies (1.12), whereCR1=−E2,k(γ+N)Dk−(N+1).$$C_{R}^{1}=-\frac{E_{2,k}}{(\gamma+N)D_{k}-(N+1)}.$$Theorem 1.2Let b satisfy (b1) -(b2) with μ ∈ (0, 1), g satisfy (g1) -(g2) with (1−μ)(γ+N)Dk>N + 1, and if (S2) holds in (g2),we further suppose that (g3) with θ > 0 and the following hold(1.15)θ ( N + 1 ) μ − N > γ > N + 1 ( 1 − μ ) D k − N , i f D k ∈ ( 0 , 1 ) , θ ( N + 1 ) − N > γ > max { N + 1 ( 1 − μ ) D k − N , 2 N + 1 N } a n d μ D k ∈ ( 0 , 1 ) , i f D k ∈ [ 1 , ∞ ) . $$\begin{equation}\begin{cases}\frac{\theta(N+1)}{\mu}-N>\gamma>\frac{N+1}{(1-\mu)D_{k}}-N,\, &\,{ if }\,D_{k}\in(0, 1),\\\theta(N+1)-N>\gamma>\max\big\{\frac{N+1}{(1-\mu)D_{k}}-N,\,\,\,\frac{2N+1}{N}\big\} \,{ and }\,\mu D_{k}\in (0, 1),\,&\,{ if }\,D_{k}\in[1, \infty).\end{cases}\end{equation}$$If k ∈ Λ1, then the unique strictly convex solution u to problem (1.1) satisfies(1.16)ξ−ψ(K(d(x)))(1+C˜−(d(x))μ+o((d(x))μ))≤−u(x)≤ξ+ψ(K(d(x)))(1+C˜+(d(x))μ+o((d(x))μ)),d(x)→0,$$\begin{equation}\begin{split}&\xi_{-}\psi(K(d(x)))\big(1+\tilde{C}_{-}(d(x))^{\mu}+o((d(x))^{\mu})\big)\\&\leq -u(x)\leq\xi_{+}\psi(K(d(x)))\big(1+\tilde{C}_{+}(d(x))^{\mu}+o((d(x))^{\mu})\big),\,d(x)\rightarrow0,\end{split}\end{equation}$$where ψ is uniquely determined by (1.4), ξ ± are given by (1.7) andC ~ ± = B 0 ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) H + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ≥ 0 , C ~ + = B 0 ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) H m ^ − m ^ + + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 < 0 , C ~ − = B 0 ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) H m ^ + m ^ − + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 < 0 $$\begin{cases}\tilde{C}_{\pm}=\frac{B_{0}(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}{\mathfrak{H}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}\geq0,\\\tilde{C}_{+}=\frac{B_{0}(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}{\mathfrak{H}\frac{\hat{m}_{-}}{\hat{m}_{+}}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}<0,\\\tilde{C}_{-}=\frac{B_{0}(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}{\mathfrak{H}\frac{\hat{m}_{+}}{\hat{m}_{-}}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}<0\end{cases}$$withH=((1−μ)(γ+N)Dk−(N+1))(N(N+1)+μ(N−1)(γ+N)Dk)+μ(γ+N)Dk((N−2)(N+1)+μ(N−1)(γ+N)Dk)+μ(1−μ)(γ+N)2Dk2.$$\begin{split}\mathfrak{H}&=\big((1-\mu)(\gamma+N)D_{k}-(N+1)\big)\big(N(N+1)+\mu(N-1)(\gamma+N)D_{k}\big)\\&+\mu(\gamma+N)D_{k}\big((N-2)(N+1)+\mu(N-1)(\gamma+N)D_{k}\big)+\mu(1-\mu)(\gamma+N)^{2}D_{k}^{2}.\end{split}$$Corollary 1.2In Theorem 1.2, if Ω is a ball with radius R and center x0, then the unique strictly convex solution u to problem (1.1) satisfies−u(x)=ξR1ψ(K(R−r))(1+CR2(R−r)μ+o((R−r)μ)),r→R,$$-u(x)=\xi_{R}^{1}\psi(K(R-r))\big(1+C_{R}^{2}(R-r)^{\mu}+o((R-r)^{\mu})\big),\,r\rightarrowR,$$whereξR1$ \xi_{R}^{1} $is given by (1.13), r=∣x − x0∣ andCR2=B0(N+1)((γ+N)Dk−(N+1))H+γ(N+1)((γ+N)Dk−(N+1)).$$C_{R}^{2}=\frac{B_{0}(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}{\mathfrak{H}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)}.$$Remark 1.1In Theorem 1.2, if we replace k ∈ Λ1 byk∈Λ′:={k∈Λ:limt→0+t−μ(ddt(K(t)k(t))−Dk)=E1,k}$$k\in\Lambda':=\bigg\{k\in\Lambda:\lim_{t\rightarrow0^{+}}t^{-\mu}\bigg(\frac{d}{dt}\bigg(\frac{K(t)}{k(t)}\bigg)-D_{k}\bigg)=E_{1,k}\bigg\}$$and other conditions still hold, then (1.16) holds, whereC ~ ± = ( N + 1 ) [ B 0 ( ( γ + N ) D k − ( N + 1 ) ) − ( γ + N ) E 1 , k ] H + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ( ( γ + N ) D k − ( N + 1 ) ) ≥ ( γ + N ) E 1 , k , C ~ + = ( N + 1 ) [ B 0 ( ( γ + N ) D k − ( N + 1 ) ) − ( γ + N ) E 1 , k ] H m ^ − m ^ + + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ( ( γ + N ) D k − ( N + 1 ) ) < ( γ + N ) E 1 , k , C ~ − = ( N + 1 ) [ B 0 ( ( γ + N ) D k − ( N + 1 ) ) − ( γ + N ) E 1 , k ] H m ^ + m ^ − + γ ( N + 1 ) ( ( γ + N ) D k − ( N + 1 ) ) , i f B 0 ( ( γ + N ) D k − ( N + 1 ) ) < ( γ + N ) E 1 , k . $$\begin{cases}\tilde{C}_{\pm}=\frac{(N+1)\big[B_{0}\big((\gamma+N)D_{k}-(N+1)\big)-(\gamma+N)E_{1,k}\big]}{\mathfrak{H}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},&\,{ if }\,B_{0}((\gamma+N)D_{k}-(N+1))\geq(\gamma+N)E_{1,k},\\\tilde{C}_{+}=\frac{(N+1)\big[B_{0}\big((\gamma+N)D_{k}-(N+1)\big)-(\gamma+N)E_{1,k}\big]}{\mathfrak{H}\frac{\hat{m}_{-}}{\hat{m}_{+}}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}((\gamma+N)D_{k}-(N+1))<(\gamma+N)E_{1,k},\\\tilde{C}_{-}=\frac{(N+1)\big[B_{0}\big((\gamma+N)D_{k}-(N+1)\big)-(\gamma+N)E_{1,k}\big]}{\mathfrak{H}\frac{\hat{m}_{+}}{\hat{m}_{-}}+\gamma(N+1)\big((\gamma+N)D_{k}-(N+1)\big)},\,&\,{ if }\,B_{0}((\gamma+N)D_{k}-(N+1))<(\gamma+N)E_{1,k}.\end{cases}$$When b is critical singular on ∂Ω, we have the following second boundary behavior.Theorem 1.3Let b satisfy (b1) and (b3), g satisfy (g1) -(g2), and if (S2) holds in (g2), we further suppose that (g3) -(g4) hold with θ = 0 in (g3) and with β ∈ (0, 1] in (g4). If L˜∈Lβ$ \tilde{L}\in \mathfrak{L}_{\beta} $with(1.17)E3<0,β∈(0,1),−1,β=1,$$\begin{equation}E_{3}<\begin{cases} 0,\,&\beta\in(0, 1),\\-1,\, &\beta=1,\end{cases}\end{equation}$$then the unique strictly convex solution u to problem (1.1) satisfiesη − ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + C − ∗ M ( d ( x ) ) + o ( M ( d ( x ) ) ) ) ≤ − u ( x ) ≤ η + ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + C + ∗ M ( d ( x ) ) + o ( M ( d ( x ) ) ) ) , d ( x ) → 0 , $$\begin{split}&\eta_{-}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+C_{-}^{*}\mathfrak{M}(d(x))+o(\mathfrak{M}(d(x)))\big)\\&\leq -u(x)\leq\eta_{+}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+C_{+}^{*}\mathfrak{M}(d(x))+o(\mathfrak{M}(d(x)))\big),\,d(x)\rightarrow0,\end{split}$$where ψ is uniquely determined by (1.4),η±=((NN+1)Nγ+Nγ−1mˆ±)−1/(γ+N),$$\begin{align}\eta_{\pm}=\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\gamma+N}{\gamma-1}\hat{m}_{\pm}\bigg)^{-1/(\gamma+N)},\end{align}$$M ( d ( x ) ) = ( − ln ⁡ ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ) − β , $$\mathfrak{M}(d(x))=\bigg(-\ln\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\bigg)^{-\beta},$$andC ± ∗ = D ± , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ = 0 , 0 , i f ( S 1 ) h o l d s , $$\begin{align}C_{\pm}^{*}=\begin{cases} \mathfrak{D}_{\pm},\,&\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})} \,{ hold\, with}\, \theta=0,\\0,\, &\,{ if }\,\mathbf{(S_{1})} \,{ holds},\end{cases}\end{align}$$whereD ± = X ± γ + N , i f X + ≥ 0 a n d X − ≥ 0 , D + = X + γ + N m ^ − m ^ + a n d D − = X − γ + N m ^ + m ^ − , i f X + ≤ 0 a n d X − ≤ 0 , D + = X + γ + N a n d D − = X − γ + N m ^ + m ^ − , i f X + > 0 a n d X − < 0 , D + = X + γ + N m ^ − m ^ + a n d D − = X − γ + N , i f X + < 0 a n d X − > 0 $$\begin{cases}\mathfrak{D}_{\pm}=\frac{\mathfrak{X_{\pm}}}{\gamma+N},\,& \,{ if }\,\mathfrak{X}_{+}\geq0 \,{ and }\,\mathfrak{X}_{-}\geq0,\\\mathfrak{D}_{+}=\frac{\mathfrak{X}_{+}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\mathfrak{D}_{-}=\frac{\mathfrak{X}_{-}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,\mathfrak{X}_{+}\leq0 \,{ and }\,\mathfrak{X}_{-}\leq0,\\\mathfrak{D}_{+}=\frac{\mathfrak{X}_{+}}{\gamma+N}\,{ and }\,\mathfrak{D}_{-}=\frac{\mathfrak{X}_{-}}{\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{-}}},\,&\,{ if }\,\mathfrak{X}_{+}>0 \,{ and }\,\mathfrak{X}_{-}<0,\\\mathfrak{D}_{+}=\frac{\mathfrak{X}_{+}}{\gamma+N\frac{\hat{m}_{-}}{\hat{m}_{+}}}\,{ and }\,\mathfrak{D}_{-}=\frac{\mathfrak{X}_{-}}{\gamma+N},\,&\,{ if }\,\mathfrak{X}_{+}<0 \,{ and }\,\mathfrak{X}_{-}>0\end{cases}$$withX±=(γ+N)−1(N+1γ−1+ln((NN+1)Nγ+Nγ−1)+lnmˆ±)σ.$$\mathfrak{X}_{\pm}=(\gamma+N)^{-1}\bigg(\frac{N+1}{\gamma-1}+\ln\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\gamma+N}{\gamma-1}\bigg)+\ln\hat{m}_{\pm}\bigg)\sigma.$$Corollary 1.3In Theorem 1.3, if Ω is a ball with radius R and center x0, then the unique strictly convex solution u to problem (1.1) satisfies− u ( x ) = η R 1 ψ ( ( ∫ 0 R − r L ( s ) s d s ) N N + 1 ) ( 1 + C R 3 M ( R − r ) + o ( M ( R − r ) ) ) , r → R , $$-u(x)=\eta_{R}^{1}\psi\bigg(\bigg(\int\limits_{0}^{R-r}\frac{L(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+C_{R}^{3}\mathfrak{M}(R-r)+o(\mathfrak{M}(R-r))\big),\,r\rightarrowR,$$whereηR1=((NN+1)Nγ+N(γ−1)RN−1)−1/(γ+N),r=|x−x0|$$\eta_{R}^{1}=\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\gamma+N}{(\gamma-1)R^{N-1}}\bigg)^{-1/(\gamma+N)},\,r=|x-x_{0}|$$andCR3=(N+1γ−1+ln((NN+1)Nγ+Nγ−1)−(N−1)lnR)σ(γ+N)2.$$C_{R}^{3}=\frac{\big(\frac{N+1}{\gamma-1}+\ln\big(\big(\frac{N}{N+1}\big)^{N}\frac{\gamma+N}{\gamma-1}\big)-(N-1)\lnR\big)\sigma}{(\gamma+N)^{2}}.$$Remark 1.2In Theorem 1.3, we obtain by Lemma 3.2 (i) that if L˜∈Lβ$ \tilde{L}\in \mathfrak{L}_{\beta} $with (1.17), then (1.3) holds. But, when L˜∈Lβ$ \tilde{L}\in\mathfrak{L}_{\beta} $with β > 1, by a direct calculation we see that(1.18)∫ 0 t 0 L ~ ( s ) s d s = ∞ . $$\begin{equation}\int\limits_{0}^{t_{0}}\frac{\tilde{L}(s)}{s}ds=\infty.\end{equation}$$Since limt→0+(−lnt)βy(t)=E3$ \lim_{t\rightarrow0^{+}}(-\ln t)^{\beta}y(t)=E_{3} $, for any ε > 0 we can choose a small enough constantt˜<min{t0,1}$ \tilde{t}<\min\{t_{0}, 1\} $such thatE3−ε(−lnt)β≤y(t),t∈(0,t˜].$$\frac{E_{3}-\varepsilon}{(-\ln t)^{\beta}}\leq y(t),\,t\in(0,\tilde{t}].$$A simple calculation shows thatexp ⁡ ( ∫ t t ~ y ( s ) s d s ) ≥ exp ⁡ ( ∫ t t ~ E 3 − ε s ( − ln ⁡ s ) β d s ) = exp ⁡ ( E 3 − ε β − 1 ( ( − ln ⁡ t ~ ) 1 − β − ( − ln ⁡ t ) 1 − β ) ) . $$\begin{align}\exp\bigg(\int\limits_{t}^{\tilde{t}}\frac{y(s)}{s}ds\bigg)\geq\exp\bigg(\int\limits_{t}^{\tilde{t}}\frac{E_{3}-\varepsilon}{s(-\lns)^{\beta}}ds\bigg)=\exp\bigg(\frac{E_{3}-\varepsilon}{\beta-1}\big((-\ln\tilde{t})^{1-\beta}-(-\ln t)^{1-\beta}\big)\bigg).\end{align}$$Moreover, we have(1.19)lim t → 0 + exp ⁡ ( ∫ t t ~ E 3 − ε s ( − ln ⁡ s ) β d s ) = lim t → 0 + exp ⁡ ( E 3 − ε β − 1 ( ( − ln ⁡ t ~ ) 1 − β − ( − ln ⁡ t ) 1 − β ) ) = exp ⁡ ( E 3 − ε β − 1 ( − ln ⁡ t ~ ) 1 − β ) > 0. $$\begin{equation}\begin{split}\lim\limits_{t\rightarrow0^{+}}\exp\bigg(\int\limits_{t}^{\tilde{t}}\frac{E_{3}-\varepsilon}{s(-\lns)^{\beta}}ds\bigg)&=\lim\limits_{t\rightarrow0^{+}}\exp\bigg(\frac{E_{3}-\varepsilon}{\beta-1}\big((-\ln\tilde{t})^{1-\beta}-(-\lnt)^{1-\beta}\big)\bigg)\\&=\exp\bigg(\frac{E_{3}-\varepsilon}{\beta-1}(-\ln\tilde{t})^{1-\beta}\bigg)>0.\end{split}\end{equation}$$It follows by the definition of L˜$ \tilde{L} $in (b3) that there exists a positive constant C˜$ \tilde{C} $such that(1.20)C ~ exp ⁡ ( ∫ t t 0 E 3 − ε s ( − ln ⁡ s ) β d s ) ≤ L ~ ( t ) , t ∈ ( 0 , t 0 ] . $$\begin{equation}\tilde{C}\exp\bigg(\int\limits_{t}^{t_{0}}\frac{E_{3}-\varepsilon}{s(-\lns)^{\beta}}ds\bigg)\leq \tilde{L}(t),\,t\in(0, t_{0}].\end{equation}$$Combining (1.19) with (1.20), we obtain (1.18) holds.Remark 1.3If y ( t ) = ± ( − ln ⁡ t ) − 1 , 0 < t ≤ t ~ < min { t 0 , 1 } $ y(t)=\pm(-\ln t)^{-1},\,0<t\leq \tilde{t}< \min\{t_{0}, 1\} $in (b3), then we haveL ~ ( t ) = exp ⁡ ( ∫ t t 0 y ( s ) s d s ) = exp ⁡ ( ∫ t ~ t 0 y ( s ) s d s ) exp ⁡ ( ± ∫ t t ~ d s s ( − ln ⁡ s ) ) = exp ⁡ ( ∫ t ~ t 0 y ( s ) s d s ) ( ln ⁡ t ln ⁡ t ~ ) ± 1 ∈ L β , $$\begin{split}\tilde{L}(t)=\exp\bigg(\int\limits_{t}^{t_{0}}\frac{y(s)}{s}ds\bigg)&=\exp\bigg(\int\limits_{\tilde{t}}^{t_{0}}\frac{y(s)}{s}ds\bigg)\exp\bigg(\pm\int\limits_{t}^{\tilde{t}}\frac{ds}{s(-\ln s)}\bigg)\\&=\exp\bigg(\int\limits_{\tilde{t}}^{t_{0}}\frac{y(s)}{s}ds\bigg)\bigg(\frac{\lnt}{\ln \tilde{t}}\bigg)^{\pm1}\in \mathfrak{L}_{\beta},\end{split}$$where β ∈ (0, 1) with E3=0 or β=1 with E3=± 1. It’s clear that (1.8) holds here. This implies that when L˜∈Lβ$ \tilde{L}\in \mathfrak{L}_{\beta} $with β ≤ 1, (1.3) is not always true if (1.17) fails.Remark 1.4In Theorem 1.3, if we substitute L˜∈Lβ0$ \tilde{L}\in \mathfrak{L}_{\beta_{0}} $(β0>β) andE3<0,β0∈(0,1),−1,β0=1,$$\begin{align}E_{3}<\begin{cases} 0,\,&\beta_{0}\in(0, 1),\\-1,\, &\beta_{0}=1,\end{cases}\end{align}$$for L˜∈Lβ$ \tilde{L}\in\mathfrak{L} _{\beta} $with (1.17), then the conclusion of Theorem 1.3 still holds.Remark 1.5L˜∈L$ \tilde{L}\in\mathcal {L} $is normalized slowly varying at zero and limt→0+tL˜′(t)L˜(t)=0$ \lim_{t\rightarrow0^{+}}\frac{t\tilde{L}'(t)}{\tilde{L}(t)}=0 $.The rest of the paper is organized as follows. In Section 2, we give some bases of Karamata regular variation theory. In Section 3, we show some auxiliary lemmas. The proofs of Theorems 1.1-1.3 are given in Section 4.2Some basic facts from Karamata regular variation theoryIn this section, we introduce some preliminaries of Karamata regular variation theory which come from [3], [32], [40]- [41].Definition 2.1A positive continuous function g defined on (0, a0], for some a0 > 0, is called regularly varying at zero with index p, denoted by g ∈ RVZp, if for each ξ>0 and some p∈R$ p \in \mathbb R $,(2.1)limt→0+g(ξt)g(t)=ξp.$$\begin{equation}\lim_{t\rightarrow 0^{+}} \frac{g(\xi t)}{g(t)}= \xi^p.\end{equation}$$In particular, when p=0, g is called slowly varying at zero.Clearly, if g ∈ RVZp, then L(t)≔ g(t)/tp is slowly varying at zero.Proposition 2.2(Uniform Convergence Theorem). If g ε RVZp, then (2.1) holds uniformlyfor ξ ε [c1, c2], where 0< c1 < c2 < a0.Proposition 2.3(Representation Theorem). A function L is slowly varying at zero if and only if it may be written in the formL ( t ) = l ( t ) e x p ∫ t a 0 y ( s ) s d s , t ≤ a 0 , $$\begin{align}L(t)=l(t) {\rm exp} \left( \int\limits_{t}^{a_{0}} \frac {y(s)}{s} ds\right), \t \leq a_0,\end{align}$$where the functions l and y are continuous and for t → 0+, y(t) → 0 and l(t) → c0 with c0 > 0.Definition 2.4The functionL ( t ) = c 0 e x p ∫ t a 0 y ( s ) s d s , t ≤ a 0 , $$L(t)=c_{0}{\rm exp} \left( \int\limits_{t}^{a_{0}} \frac {y(s)}{s} ds\right), \t \leq a_0,$$is called normalized slowly varying at zero andh(t)=tpL(t), t≥a0$$\begin{align}h(t)=t^p L(t), \t \geq a_0\end{align}$$is called normalized regularly varying at zero with index ρ (written as f ∈ NRVZp).A function h ε C1(0, a0] for some a0 > 0 belongs to NRVZp if and only iflimt→0+th′(t)h(t)=p.$$\begin{align}\lim_{t\rightarrow0^{+}}\frac{th'(t)}{h(t)}=p.\end{align}$$Proposition 2.5Let functions L, L1 be slowly varying at zero, then(i) Lp, p∈R$ p\in\mathbb{R} $, L1· L and L1 ∘ L satisfying limt→0+L(t)=0$ \lim_{t\rightarrow0^{+}}L(t)=0 $, are also slowly varying at zero;(ii) for any p > 0,tpL(t)→0 and t−pL(t)→∞ ast→0+;$$\begin{align}t^{p}L(t)\rightarrow 0 \mbox{ and }t^{-p}L(t)\rightarrow\infty\mbox{ as}t\rightarrow0^{+};\end{align}$$(iii) for any p ∈ ℝ,ln ⁡ L ( t ) / ln ⁡ t → 0 and ln ⁡ ( t p L ( t ) ) / ln ⁡ t = p as t → 0 + . $$\begin{align}\ln L(t)/\ln t\rightarrow0 \mbox{ and }\ln(t^{p}L(t))/\ln t=p \mbox{ as}\,t\rightarrow0^{+}.\end{align}$$Proposition 2.6Let g1∈(N)RVZp1 and g2∈(N)RVZp2, then g1⋅g2∈(N)RVZp1+p2.$ g_{1}\in (N)RVZ_{p_{1}}\text{ and }g_{2}\in (N)RVZ_{p_{2}}\text{, then }g_{1}\cdot g_{2}\in (N)RVZ_{p_{1}+p_{2}}. $Proposition 2.7Let g1∈(N)RVZp1,g2∈(N)RVZp2 and limt→0+g2(t)=0, then g1∘g2∈(N)RVZp1p2.$ g_{1}\in (N)RVZ_{p_{1}},\,\,g_{2}\in (N)RVZ_{p_{2}}\text{ and }\lim_{t\rightarrow0^{+}}g_{2}(t)=0\text{, then }g_{1}\circ g_{2}\in(N)RVZ_{p_{1}p_{2}}. $Proposition 2.8(Asymptotic Behavior). Let L is a slowly varying functionat zero, then for a1 > 0,(i) ∫ta1spL(s)ds∼−(1+p)−1t1+pL(t), forp<−1,t→0+;$ \int_{t}^{a_{1}}s^{p}L(s)ds\sim -(1+p)^{-1}t^{1+p}L(t),\mbox{ for} p < -1,\,t\rightarrow 0^{+}; $(ii)∫0tspL(s)ds∼(1+p)−1t1+pL(t), forp>−1,t→0+.$ \int_{0}^{t}s^{p}L(s)ds \sim (1+p)^{-1}t^{1+p}L(t), \mbox{ for}p>-1,\, t\rightarrow0^{+}. $3Auxiliary resultsIn this section, we show some useful results, which are necessary in the proofs of our results.Lemma 3.1(Lemma 2.9 in [51]). Let k ∈ Λ, thenwhen k is non-decreasing, Dk ∈ [0, 1]; and when k is non-increasing, Dk ∈ [1, ∞);limt→0+K(t)k(t)=0$ \lim_{t\rightarrow0^{+}}\frac{K(t)}{k(t)}=0 $, limt→0+K(t)tk(t)=Dk$ \lim_{t\rightarrow0^{+}}\frac{K(t)}{tk(t)}=D_{k} $and limt→0+K(t)k′(t)k2(t)=1−Dk;$ \lim_{t\rightarrow0^{+}}\frac{K(t)k'(t)}{k^{2}(t)}=1-D_{k}; $when Dk > 0 and Dk≠1, k∈NRVZ(1−Dk)/Dk;$ k\in NRVZ_{(1-D_{k})/D_{k}}; $when Dk=1, k is normalized slowlyvarying at zero.Lemma 3.2Let L˜∈Lβ$ \tilde{L}\in\mathfrak{L}_{\beta} $and (1.17) hold, then∫0tL˜(s)sds<∞;$ \int_{0}^{t}\frac{\tilde{L}(s)}{s}ds<\infty; $limt→0+(−lnt)βL˜(s)∫0tL˜(s)sds=−E3,β∈(0,1),−(E3+1),β=1;$\lim_{t\rightarrow0^{+}}(-\lnt)^{\beta}\frac{\tilde{L}(s)}{\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds}=\begin{cases}-E_{3},\, &\beta\in(0, 1),\\-(E_{3}+1),\, &\beta=1;\end{cases}$limt→0+(−lnt)βL˜′(t)tL˜(t)=E3.$\lim_{t\rightarrow0^{+}}(-\lnt)^{\beta}\frac{\tilde{L}'(t)t}{\tilde{L}(t)}= E_{3}. $Proof. (i) Bylimt→0+(−lnt)βy(t)=E3<0,β∈(0,1),−1,β=1,$$\lim\limits_{t \rightarrow 0^{+}}(-\ln t)^{\beta} y(t)=E_{3}<\left\{\begin{array}{ll}0, & \beta \in(0,1), \\ -1, & \beta=1,\end{array}\right.$$we see that for any 0<ε<−E3+τ2$0<\varepsilon<-\frac{E_{3}+\tau}{2} $withτ=0,β∈(0,1),1,β=1,$$\tau=\left\{\begin{array}{ll}0, & \beta \in(0,1), \\ 1, & \beta=1,\end{array}\right.$$there exists a small enough positive constant t* ∈ (0, 1) such thaty(t)≤E3+ε(−lnt)β,t∈(0,t∗] with E3+ε<0,β∈(0,1),−1,β=1.$$y(t)\leq \frac{E_{3}+\varepsilon}{(-\ln t)^{\beta}},\,t\in(0, t_{*}]\mbox{ with }E_{3}+\varepsilon<\begin{cases} 0,\,&\beta\in (0, 1),\\-1,\, &\beta=1.\end{cases}$$A straightforward calculation shows that for any t∈0,t⋆,$t \in\left(0, t_{\star}\right],$exp(∫tt∗y(s)sds)≤exp(∫tt∗E3+εs(−lns)βds)=exp(E3+ε1−β((−lnt)1−β−(−lnt∗)1−β)),β∈(0,1),(lntlnt∗)E3+ε,β=1.$$\begin{split}\exp\bigg(\int\limits_{t}^{t_{*}}\frac{y(s)}{s}ds\bigg)&\leq\exp\bigg(\int\limits_{t}^{t_{*}}\frac{E_{3}+\varepsilon}{s(-\ln s)^{\beta}}ds\bigg)\\&=\begin{cases}\exp\bigg(\frac{E_{3}+\varepsilon}{1-\beta}\big((-\lnt)^{1-\beta}-(-\ln t_{*})^{1-\beta}\big)\bigg),\,&\beta\in(0, 1),\\\big(\frac{\ln t}{\ln t_{*}}\big)^{E_{3}+\varepsilon},\, &\beta=1.\end{cases}\end{split}$$So, we see that there exists a large constant C > 0 such that for any t∈(0,t∗],$t\in(0, t_{*}],$(3.1)L˜(t)≤Cexp(E3+ε1−β(−lnt)1−β),β∈(0,1),C(−lnt)E3+ε,β=1.$$\begin{equation}\tilde{L}(t)\leq \begin{cases}C\exp\big(\frac{E_{3}+\varepsilon}{1-\beta}(-\lnt)^{1-\beta}\big),\,&\beta\in(0, 1),\\C(-\ln t)^{E_{3}+\varepsilon},\, &\beta=1.\end{cases}\end{equation}$$Case 1. If β∈(0, 1), it is clear that thatexp(−E3+ε1−β(−lnt)1−β)=∑n=0∞(−E3+ε1−β)n(−lnt)n(1−β)n!.$$\exp\bigg(-\frac{E_{3}+\varepsilon}{1-\beta}(-\lnt)^{1-\beta}\bigg)=\sum\limits_{n=0}^{\infty}\frac{\big(-\frac{E_{3}+\varepsilon}{1-\beta}\big)^{n}(-\lnt)^{n(1-\beta)}}{n!}.$$This together with (3.1) implies that we can choose a positive integer n* >(1-β)-1 such that(3.2)Cexp(E3+ε1−β(−lnt)1−β)=C(exp(−E3+ε1−β(−lnt)1−β))−1=C(∑n=0∞(−E3+ε1−β)n(−lnt)n(1−β)n!)−1≤C((−E3+ε1−β)n∗(−lnt)n∗(1−β))−1=C1(−lnt)−n∗(1−β),t∈(0,t∗],$$\begin{equation}\label{dll}\begin{split}C\exp\bigg(\frac{E_{3}+\varepsilon}{1-\beta}(-\lnt)^{1-\beta}\bigg)&=C\bigg(\exp\bigg(-\frac{E_{3}+\varepsilon}{1-\beta}(-\lnt)^{1-\beta}\bigg)\bigg)^{-1}\\&=C\bigg(\sum_{n=0}^{\infty}\frac{\big(-\frac{E_{3}+\varepsilon}{1-\beta}\big)^{n}(-\lnt)^{n(1-\beta)}}{n!}\bigg)^{-1}\\&\leqC\bigg(\bigg(-\frac{E_{3}+\varepsilon}{1-\beta}\bigg)^{n_{*}}(-\lnt)^{n_{*}(1-\beta)}\bigg)^{-1}\\&=C_{1}(-\ln t)^{-n_{*}(1-\beta)},\,t\in(0, t_{*}],\end{split}\end{equation}$$where C1=C(−E3+ε1−β)−n∗$C_{1}=C\big(-\frac{E_{3}+\varepsilon}{1-\beta}\big)^{-n_{*}}$. So, we have∫0t∗L˜(s)sds≤C∫0t∗exp(E3+ε1−β(−lns)1−β)sds≤C1∫0t∗dss(−lns)n∗(1−β)=C1(−lnt∗)1−n∗(1−β)n∗(1−β)−1<∞.$$\begin{split}\int\limits_{0}^{t_{*}}\frac{\tilde{L}(s)}{s}ds &\leqC\int\limits_{0}^{t_{*}}\frac{\exp\big(\frac{E_{3}+\varepsilon}{1-\beta}(-\ln s)^{1-\beta}\big)}{s}ds\\&\leq C_{1}\int\limits_{0}^{t_{*}}\frac{ds}{s (-\ln s)^{n_{*}(1-\beta)}}=\frac{C_{1}(-\lnt_{*})^{1-n_{*}(1-\beta)}}{n_{*}(1-\beta)-1}<\infty.\end{split}$$Case 2. If β=1, then by (3.1) we obtain∫0t∗L˜(s)sds≤∫0t∗C(−lns)E3+εsds=−C(−lnt∗)1+E3+ε1+E3+ε<∞.$$\int\limits_{0}^{t_{*}}\frac{\tilde{L}(s)}{s}ds\leq\int\limits_{0}^{t_{*}}\frac{C(-\lns)^{E_{3}+\varepsilon}}{s}ds=-\frac{C(-\lnt_{*})^{1+E_{3}+\varepsilon}}{1+E_{3}+\varepsilon}<\infty.$$Combining Cases 1-2, we obtain (i) holds.(ii) It follows by (3.1)-(3.2) that limt→0+(−lnt)βL˜(t)=0.$\lim_{t\rightarrow0^{+}}(-\ln t)^{\beta}\tilde{L}(t)=0.$By the l'Hospital’s rule, we obtainlimt→0+(−lnt)βL˜(s)∫0tL˜(s)sds=limt→0+(−(−lnt)βy(t)−β(−lnt)β−1)=−E3,β∈(0,1),−(E3+1),β=1.$$\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}\frac{\tilde{L}(s)}{\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds}=\lim\limits_{t\rightarrow0^{+}}\big(-(-\lnt)^{\beta}y(t)-\beta(-\ln t)^{\beta-1}\big)=\begin{cases}-E_{3},\, &\beta\in(0, 1),\\-(E_{3}+1),\, &\beta=1.\end{cases}$$(iii) A direct calculation shows thatlimt→0+(−lnt)βL˜′(t)tL˜(t)=limt→0+(−lnt)βy(t)=E3.$$\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}\frac{\tilde{L}'(t)t}{\tilde{L}(t)}=\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}y(t)=E_{3}.$$□Define(3.3)φ(t):=∫0t((N+1)G(s))−1/(N+1)ds,t>0,$$\begin{equation}\varphi(t):=\int\limits_{0}^{t}((N+1)G(s))^{-1/(N+1)}ds,\,t>0,\end{equation}$$where φ is the inverse of solution ψ to (1.4).Lemma 3.3Let g satisfy (g1)-(g2), thenlimt→0+tg′(t)g(t)=−γ;$\lim_{t\rightarrow0^{+}}\frac{tg'(t)}{g(t)}=-\gamma;$G(t)∼t1−γγ−1Lˆ0(t),t→0+;$G(t)\sim\frac{t^{1-\gamma}}{\gamma-1}\hat{L}_{0}(t),\,t\rightarrow0^{+};$φ(t)∼(γ−1N+1)1/(N+1)N+1γ+Ntγ+NN+1(Lˆ0(t))−1/(N+1),t→0+,$\varphi(t)\sim\big(\frac{\gamma-1}{N+1}\big)^{1/(N+1)}\frac{N+1}{\gamma+N}t^{\frac{\gamma+N}{N+1}}(\hat{L}_{0}(t))^{-1/(N+1)},\,t\rightarrow0^{+},$ whereLˆ0(t)=c0exp(∫ta1f(s)sds),c0=g(a1)a1γ;$$\hat{L}_{0}(t)=c_{0}\exp\bigg(\int\limits_{t}^{a_{1}}\frac{f(s)}{s}ds\bigg),\,c_{0}=g(a_{1})a_{1}^{\gamma};$$limt→0+φ(t)tφ′(t)=limt→0+φ(t)((N+1)G(t))1/(N+1)t=N+1γ+N,i.e., φ∈NRVZγ+NN+1;$\lim_{t\rightarrow0^{+}}\frac{\varphi(t)}{t\varphi'(t)}=\lim_{t\rightarrow0^{+}}\frac{\varphi(t)((N+1)G(t))^{1/(N+1)}}{t}=\frac{N+1}{\gamma+N},\mbox{i.e., }\varphi\inNRVZ_{\frac{\gamma+N}{N+1}};$limt→0+((N+1)G(t))N/(N+1)g(t)φ(t)=γ+Nγ−1.$\lim_{t\rightarrow0^{+}}\frac{((N+1)G(t))^{N/(N+1)}}{g(t)\varphi(t)}=\frac{\gamma+N}{\gamma-1}.$proof. (i) It follows by (g2) that (i) holds.(ii)-(iii) By (g2) we see that(3.4)g(t)=t−γLˆ0(t),t∈(0,a1].$$\begin{equation}g(t)=t^{-\gamma}\hat{L}_{0}(t),\, t\in(0, a_{1}].\end{equation}$$It follows by Proposition 2.8 (i) that (ii) holds. Furthermore, we have((N+1)G(t))−1/(N+1)∼(γ−1N+1)1/(N+1)t(γ−1)/(N+1)(Lˆ0(t))−1/(N+1),t→0+.$$((N+1)G(t))^{-1/(N+1)}\sim\bigg(\frac{\gamma-1}{N+1}\bigg)^{1/(N+1)}t^{(\gamma-1)/(N+1)}(\hat{L}_{0}(t))^{-1/(N+1)},\,t\rightarrow0^{+}.$$It follows by Proposition 2.8 (ii) that (iii) holds.(iv)-(v) (3.3) implies that(3.5)φ′(t)=((N+1)G(t))−1/(N+1),t>0.$$\begin{equation}\varphi'(t)=((N+1)G(t))^{-1/(N+1)},\,t>0.\end{equation}$$By (3.4)-(3.5) and (i)-(ii), we obtain (iv)-(v) hold.Lemma 3.4Suppose that g satisfies (g1)-(g2). In particular, if (S2) holds in (g2), we suppose that (g3) holds. And if θ = 0 in (g3), we further suppose that (g4) holds. Then,limt→0+υ(t)(tg′(t)g(t)+γ)=χ1,$$\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=\chi_{1},$$whereχ1=−σ,if(S2)and(g3)−(g4)holdwithθ=0in(g3),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ>0),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ(N+1)>(γ+N)ρ),υ(t)=(φ(t))−ρwithρ∈(0,1];$$\chi_{1}=\begin{cases}-\sigma, &\,{ if }\mathbf{(S_{2})} \,{ and } \,\mathbf{(g_{3})}\,{-}\mathbf{(g_{4})} \,{ hold\, with }\, \theta=0 \,{ in }\,\mathbf{(g_{3})}, \,\upsilon(t)=(-\ln t)^{\beta},\\0, &\,{ if }\,\mathbf{(S_{1})} \,{ holds\, (or}\,\mathbf{(S_{2})}\,{ and }\, \mathbf{(g_{3})} \,{ hold\, with }\,\theta>0),\, \upsilon(t)=(-\ln t)^{\beta},\\0, &\,{ if }\mathbf{(S_{1})} \,{ holds\, (or}\,\mathbf{(S_{2})}\,{ and } \mathbf{(g_{3})} \,{ hold\, with }\,\theta(N+1)>(\gamma+N)\rho),\\&\upsilon(t)=(\varphi(t))^{-\rho}\,{ with }\,\rho\in(0,1];\\\end{cases}$$limt→0+υ(t)(G(t)tg(t)−1γ−1)=χ2,$$\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)=\chi_{2},$$whereχ2=−σ(γ−1)2,if(S2)and(g3)−(g4)holdwithθ=0in(g3),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ>0),υ(t)=(−lnt)β,0,if(S1)holdsandγ>(1+ρ)N+1N+1−ρin(g2),υ(t)=(φ(t))−ρ,ρ∈(0,1],0,if(S2)and(g3)holdwithθ(N+1)ρ−N>γ>(1+ρ)N+1N+1−ρ,υ(t)=(φ(t))−ρwithρ∈(0,1];$$\chi_{2}=\begin{cases}-\frac{\sigma}{(\gamma-1)^{2}}, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\, \mathbf{(g_{3})}\,{-}\mathbf{(g_{4})} \,{ hold\, with }\, \theta=0 \,{ in }\,\mathbf{(g_{3})},\,\upsilon(t)=(-\ln t)^{\beta},\\0, &\,{ if }\mathbf{(S_{1})} \,{ holds\, (or\,}\mathbf{(S_{2})}\,{ and }\, \mathbf{(g_{3})} \,{ hold \,with}\,\theta>0), \,\upsilon(t)=(-\lnt)^{\beta},\\0, &\,{ if }\,\mathbf{(S_{1})}\,{ holds\, and }\,\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \,{ in }\,\mathbf{(g_{2})},\,\upsilon(t)=(\varphi(t))^{-\rho},\, \rho\in(0, 1],\\0, &\,{ if }\,\mathbf{(S_{2})}\,{ and }\, \mathbf{(g_{3})} \,{hold\, with}\,\frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho},\\&\upsilon(t)=(\varphi(t))^{-\rho}\,{ with }\, \rho\in(0, 1];\end{cases}$$limt→0+υ(t)(((N+1)G(t))N/(N+1)g(t)φ(t)−γ+Nγ−1)=χ3,$$\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{((N+1)G(t))^{N/(N+1)}}{g(t)\varphi(t)}-\frac{\gamma+N}{\gamma-1}\bigg)=\chi_{3},$$whereχ3=−(N+1)σ(γ−1)2,if(S2)and(g3)−(g4)holdwithθ=0in(g3),υ(t)=(−lnt)β,0,if(S1)holds(or(S2)and(g3)holdwithθ>0),υ(t)=(−lnt)β,0,if(S1)holdsandγ>(1+ρ)N+1N+1−ρin(g2),υ(t)=(φ(t))−ρ,ρ∈(0,1],0,if(S2)and(g3)holdwithθ(N+1)ρ−N>γ>(1+ρ)N+1N+1−ρ,υ(t)=(φ(t))−ρwithρ∈(0,1];$$\chi_{3}=\begin{cases}-\frac{(N+1)\sigma}{(\gamma-1)^{2}}, &\,{ if }\,\mathbf{(S_{2})} \,{ and } \,\mathbf{(g_{3})}\,{-}\,\mathbf{(g_{4})} \,{ hold \,with }\, \theta=0 \,{ in }\,\mathbf{(g_{3})}, \,\upsilon(t)=(-\ln t)^{\beta},\\0, &\,{ if }\,\mathbf{(S_{1})} \,{ holds\, (or}\,\mathbf{(S_{2})}\,{ and } \,\mathbf{(g_{3})} \,{ hold\, with}\,\theta>0), \,\upsilon(t)=(-\lnt)^{\beta},\\0, &\,{ if }\,\mathbf{(S_{1})}\,{ holds\, and}\,\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \,{ in }\,\mathbf{(g_{2})}, \,\upsilon(t)=(\varphi(t))^{-\rho},\,\rho\in(0, 1],\\0, &\,{ if }\,\mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})} \,{ hold \,with }\,\frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho},\\&\upsilon(t)=(\varphi(t))^{-\rho}\,{ with }\,\rho\in(0, 1];\end{cases}$$let ξ be a positive constant, thenlimt→0+υ(t)g(ξt)ξNg(t)−ξ−(γ+N)=χ4,$$\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\left(\frac{g(\xi t)}{\xi^{N}g(t)}-\xi^{-(\gamma+N)}\right)=\chi_{4},$$whereχ4=−σlnξξγ+N,ifS2andg3−g4holdwithθ=0ing3,v(t)=(−lnt)β,0,ifS1holdsorS2andg3holdwithθ>0,v(t)=(−lnt)β,0,ifS1holdsorS2andg3holdwithθ(N+1)>(γ+N)ρ,v(t)=(φ(t))−ρwithρ∈(0,1].$$\chi_{4}=\,\left\{\begin{array}{ll}-\frac{\sigma \ln \xi}{\xi^{\gamma+N}}, & \, { if }\,\left(\mathbf{S}_{\mathbf{2}}\right) \, { and }\,\left(\mathbf{g}_{3}\right)-\,\left(\mathbf{g}_{4}\right) \, { hold\, with } \,\theta=0 \, { in }\,\left(\mathbf{g}_{3}\right), v(t)=(-\ln t)^{\beta}, \\ 0, & \, { if }\,\left(\mathbf{S}_{1}\right) \, { holds }\,\left(\, { or }\,\left(\mathbf{S}_{\mathbf{2}}\right) \, { and }\,\left(\mathbf{g}_{3}\right) \, { hold\, with }\, \theta>0\right), v(t)=(-\ln t)^{\beta}, \\ 0, & \, { if }\,\left(\mathbf{S}_{1}\right) \, { holds }\,\left(\, { or }\,\left(\mathbf{S}_{2}\right) \, { and }\,\left(\mathbf{g}_{3}\right) \, { hold\, with }\, \theta(N+1)>(\gamma+N) \rho\right), \\ & v(t)=(\varphi(t))^{-\rho} \, { with }\, \rho \in(0,1] .\end{array}\right.$$Proof. (i) When (S2) and (g3)-(g4) hold with θ = 0 in (g3), we have(3.6)limt→0+(−lnt)β(tg′(t)g(t)+γ)=−limt→0+(−lnt)βf(t)=−σ.$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=-\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}f(t)=-\sigma.\end{equation}$$When (S1) holds, we havetg′(t)g(t)=−γ,t∈(0,a1].$$\frac{tg'(t)}{g(t)}=-\gamma,\,t\in(0, a_{1}].$$This implies that for υ (t)=(-ln t)β orυ (t)=(φ (t))-ρ, we obtain(3.7)limt→0+υ(t)(tg′(t)g(t)+γ)=0.$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=0.\end{equation}$$When (S2) and (g3)-(g4) hold with θ > 0, we conclude by Proposition 2.5 (ii) that(3.8)limt→0+(−lnt)β(tg′(t)g(t)+γ)=−limt→0+(−lnt)βf(t)=−limt→0+tθ(−lnt)βL(t)=0,$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=-\lim\limits_{t\rightarrow0^{+}}(-\lnt)^{\beta}f(t)=-\lim\limits_{t\rightarrow0^{+}}t^{\theta}(-\lnt)^{\beta}L(t)=0,\end{equation}$$where L ∈ NRVZ0. Moreover, it follows by Lemma (3.3) (iv) and Proposition (2.7) that(3.9)1/φρ∈NRVZ−(γ+N)ρN+1.$$\begin{equation}1/\varphi^{\rho}\in NRVZ_{-\frac{(\gamma+N)\rho}{N+1}}.\end{equation}$$When (g3) holds with θ (N + 1) > (γ + N)ρ, we conclude by Proposition 2.6 and Proposition 2.5 (ii) that(3.10)limt→0+(φ(t))−ρ(tg′(t)g(t)+γ)=−limt→0+(φ(t))−ρf(t)=−limt→0+tθ−(γ+N)ρN+1L(t)=0,$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}(\varphi(t))^{-\rho}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)=-\lim\limits_{t\rightarrow0^{+}}(\varphi(t))^{-\rho}f(t)=-\lim\limits_{t\rightarrow0^{+}}t^{\theta-\frac{(\gamma+N)\rho}{N+1}}L(t)=0,\end{equation}$$where L ∈ NRVZ0. So, (i) follows by (3.6)-(3.10).(ii) By (g2), we have−tg′(t)=γg(t)+g(t)f(t),t∈(0,a1].$$-tg'(t)=\gamma g(t)+g(t)f(t),\,t\in(0, a_{1}].$$Integrating it from t to a1 and integration by parts, we obtaintg(t)=(γ−1)G(t)+∫ta1g(s)f(s)ds+c,t∈(0,a1],$$tg(t)=(\gamma-1)G(t)+\int\limits_{t}^{a_{1}}g(s)f(s)ds+c,\, t\in(0, a_{1}],$$i.e.,(3.11)G(t)tg(t)−1γ−1=−1γ−1∫ta1g(s)f(s)dstg(t)−c(γ−1)g(t)t,t∈(0,a1],$$\begin{equation}\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}=-\frac{1}{\gamma-1}\frac{\int_{t}^{a_{1}}g(s)f(s)ds}{tg(t)}-\frac{c}{(\gamma-1)g(t)t},\,t\in(0,a_{1}],\end{equation}$$where c is a constant. The condition (g2) implies that g ∈ NRVZ-γ with γ > 1. Moreover, by t ↦ (-ln t)β ∈ NRVZ0 and (3.9), we see thatυ∈NRVZ0, if υ(t)=(−lnt)β,NRVZ−(γ+N)ρN+1, if υ(t)=(φ(t))−ρ.$$\upsilon\in\begin{cases}NRVZ_{0},\, &\mbox{ if }\upsilon(t)=(-\ln t)^{\beta},\\NRVZ_{-\frac{(\gamma+N)\rho}{N+1}},\,&\mbox{ if }\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$We conclude by Proposition 2.7 and Proposition 2.6 thatt↦tg(t)(υ(t))−1∈NRVZ1−γ, if υ(t)=(−lnt)β,NRVZ(1−γ)(N+1)+(γ+N)ρN+1, if υ(t)=(φ(t))−ρ.$$t\mapsto tg(t)(\upsilon(t))^{-1}\in\begin{cases}NRVZ_{1-\gamma},\, &\mbox{ if }\upsilon(t)=(-\ln t)^{\beta},\\NRVZ_{\frac{(1-\gamma)(N+1)+(\gamma+N)\rho}{N+1}},\,&\mbox{ if}\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$So, we can take L ∈ NRVZ0 such thattg(t)(υ(t))−1=tςL(t)$$tg(t)(\upsilon(t))^{-1}=t^{\varsigma}L(t)$$withς=1−γ, if υ(t)=(−lnt)β,(1−γ)(N+1)+(γ+N)ρN+1, if υ(t)=(φ(t))−ρ.$$\varsigma=\begin{cases}1-\gamma,\, &\mbox{ if }\upsilon(t)=(-\ln t)^{\beta},\\\frac{(1-\gamma)(N+1)+(\gamma+N)\rho}{N+1},\,&\mbox{ if}\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$It’s clear thatς<0, if γ>1 in (g2) and υ(t)=(−lnt)β,ς<0, if γ>(1+ρ)N+1N+1−ρ in (g2) and υ(t)=(φ(t))−ρ.$$\begin{cases}\varsigma<0,\, &\mbox{ if }\gamma>1 \mbox{ in }\mathbf{(g_{2})} \mbox{ and }\upsilon(t)=(-\ln t)^{\beta},\\\varsigma<0,\,&\mbox{ if }\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \mbox{in }\mathbf{(g_{2})}\mbox{ and }\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$By Proposition 2.5 (ii), we havelimt→0+tg(t)(υ(t))−1=∞,$$\lim\limits_{t\rightarrow0^{+}}tg(t)(\upsilon(t))^{-1}=\infty,$$i.e.,(3.12)limt→0+−c(γ−1)tg(t)(υ(t))−1=0.$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}-\frac{c}{(\gamma-1)tg(t)(\upsilon(t))^{-1}}=0.\end{equation}$$Combining with (3.6)-(3.10), by the l'Hospital’s rule, we can obtain(3.13)limt→0+υ(t)−∫ta1g(s)f(s)ds(γ−1)tg(t)=limt→0+∫a1tg(s)f(s)ds(γ−1)tg(t)(υ(t))−1=limt→0+1γ−1g(t)f(t)g(t)(υ(t))−1+tg′(t)(υ(t))−1+tg(t)[(υ(t))−1]′=limt→0+1γ−1υ(t)f(t)1+tg′(t)g(t)+t[(υ(t))−1]′(υ(t))−1=limt→0+(−lns)βf(s)−(γ−1)2, if γ>1 in (g2) and υ(t)=(−lnt)βlimt→0+(N+1)(φ(t))−ρf(t)(γ−1)((1−γ)(N+1)+(γ+N)ρ), if γ>(1+ρ)N+1N+1−ρ in (g2) and υ(t)=(φ(t))−ρ=χ2.$$\begin{equation}\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\frac{-\int_{t}^{a_{1}}g(s)f(s)ds}{(\gamma-1)tg(t)}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\int_{a_{1}}^{t}g(s)f(s)ds}{(\gamma-1)tg(t)(\upsilon(t))^{-1}}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{1}{\gamma-1}\frac{g(t)f(t)}{g(t)(\upsilon(t))^{-1}+tg'(t)(\upsilon(t))^{-1}+tg(t)[(\upsilon(t))^{-1}]'}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{1}{\gamma-1}\frac{\upsilon(t)f(t)}{1+\frac{tg'(t)}{g(t)}+\frac{t[(\upsilon(t))^{-1}]'}{(\upsilon(t))^{-1}}}\\=&\begin{cases}\lim\limits_{t\rightarrow0^{+}}\frac{(-\ln s)^{\beta}f(s)}{-(\gamma-1)^{2}},\, &\mbox{ if }\gamma>1 \mbox{ in }\mathbf{(g_{2})} \mbox{ and }\upsilon(t)=(-\ln t)^{\beta}\\\lim\limits_{t\rightarrow0^{+}}\frac{(N+1)(\varphi(t))^{-\rho}f(t)}{(\gamma-1)((1-\gamma)(N+1)+(\gamma+N)\rho)},\,&\mbox{if }\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \mbox{ in}\mathbf{(g_{2})}\mbox{ and }\upsilon(t)=(\varphi(t))^{-\rho}\end{cases}\\=&\chi_{2}.\end{split}\end{equation}$$(3.11)-(3.13) imply thatlimt→0+υ(t)G(t)tg(t)−1γ−1=−limt→0+∫ta1g(s)f(s)ds(γ−1)tg(t)(υ(t))−1−limt→0+c(γ−1)tg(t)(υ(t))−1=χ2.$$\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\left(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\right)\\=-&\lim\limits_{t\rightarrow0^{+}}\frac{\int_{t}^{a_{1}}g(s)f(s)ds}{(\gamma-1)tg(t)(\upsilon(t))^{-1}}-\lim\limits_{t\rightarrow0^{+}}\frac{c}{(\gamma-1)tg(t)(\upsilon(t))^{-1}}=\chi_{2}.\\\end{split}$$(iii) By Lemma 3.3 (iv), Proposition 2.7 and Proposition 2.6, we havet↦(υ(t))−1φ(t)∈NRVZ(γ+N)ρN+1, if υ(t)=(−lnt)β,NRVZ(γ+N)(ρ+1)N+1, if υ(t)=(φ(t))−ρ.$$t\mapsto(\upsilon(t))^{-1}\varphi(t)\in\begin{cases}NRVZ_{\frac{(\gamma+N)\rho}{N+1}},\, &\mbox{ if } \upsilon(t)=(-\ln t)^{\beta},\\NRVZ_{\frac{(\gamma+N)(\rho+1)}{N+1}},\,&\mbox{ if}\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}$$This implies that(υ(t))−1φ(t)→0 as t→0+.$$(\upsilon(t))^{-1}\varphi(t)\rightarrow0 \mbox{as }t\rightarrow0^{+}.$$Moreover, it follows by Lemma 3.3 (v) that((N+1)G(t))1/(N+1)g(t)−γ+Nγ−1φ(t)→0 as t→0+.$$\frac{((N+1)G(t))^{1/(N+1)}}{g(t)}-\frac{\gamma+N}{\gamma-1}\varphi(t)\rightarrow0\mbox{ as }t\rightarrow0^{+}.$$If ν(t) = (− ln t)β, then we have by the l’Hospital’s rule, (3.5) and (i)-(ii) that(3.14)limt→0+υ(t)(((N+1)G(t))N/(N+1)g(t)φ(t)−γ+Nγ−1)=limt→0+((N+1)G(t))N/(N+1)g(t)−γ+Nγ−1φ(t)(−lnt)−βφ(t)=limt→0+−N((N+1)G(t))−1/(N+1)g2(t)−((N+1)G(t))N/(N+1)g′(t)g2(t)−γ+Nγ−1φ′(t)(−lnt)−βφ′(t)+β(−lnt)−β−1φ(t)t−1=limt→0+−(N+1)(−lnt)β(G(t)tg(t)tg′(t)g(t)+γγ−1)=limt→0+−(N+1)(−lnt)β[(G(t)tg(t)−1γ−1)(tg′(t)g(t)+γ)−γ(G(t)tg(t)−1γ−1)+1γ−1(tg′(t)g(t)+γ)]=−(N+1)σ(γ−1)2, if (S2) and (g3)-(g4) hold with θ=0 in (g3),0, if (S1) holds (or (S2) and (g3) hold with θ>0).$$\begin{equation}\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{((N+1)G(t))^{N/(N+1)}}{g(t)\varphi(t)}-\frac{\gamma+N}{\gamma-1}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\frac{((N+1)G(t))^{N/(N+1)}}{g(t)}-\frac{\gamma+N}{\gamma-1}\varphi(t)}{(-\lnt)^{-\beta}\varphi(t)}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\frac{-N((N+1)G(t))^{-1/(N+1)}g^{2}(t)-((N+1)G(t))^{N/(N+1)}g'(t)}{g^{2}(t)}-\frac{\gamma+N}{\gamma-1}\varphi'(t)}{(-\ln t)^{-\beta}\varphi'(t)+\beta(-\ln t)^{-\beta-1}\varphi(t)t^{-1}}\\=&\lim\limits_{t\rightarrow0^{+}}-(N+1)(-\lnt)^{\beta}\bigg(\frac{G(t)}{tg(t)}\frac{tg'(t)}{g(t)}+\frac{\gamma}{\gamma-1}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}-(N+1)(-\ln t)^{\beta}\bigg[\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)-\gamma\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)\\&+\frac{1}{\gamma-1}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)\bigg]=\begin{cases}-\frac{(N+1)\sigma}{(\gamma-1)^{2}},\, &\mbox{ if }\mathbf{(S_{2})}\mbox{ and } \mathbf{(g_{3})}\mbox{-}\mathbf{(g_{4})} \mbox{ hold with } \theta=0 \mbox{ in }\mathbf{(g_{3})},\\0,\, &\mbox{ if }\mathbf{(S_{1})} \mbox{ holds (or } \mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with } \theta>0).\\\end{cases}\end{split}\end{equation}$$If ν(t) = (φ(t))−ρ, then we have by the l'Hospital’s rule, (3.5) and (i)-(ii) that(3.15)limt→0+υ(t)(((N+1)G(t))N/(N+1)g(t)φ(t)−γ+Nγ−1)=limt→0+((N+1)G(t))N/(N+1)g(t)−γ+Nγ−1φ(t)(φ(t))ρ+1=limt→0+−N((N+1)G(t))−1/(N+1)g2(t)−((N+1)G(t))N/(N+1)g′(t)g2(t)−γ+Nγ−1φ′(t)(ρ+1)(φ(t))ρφ′(t)=limt→0+−N+1ρ+1(φ(t))−ρ(G(t)tg(t)tg′(t)g(t)+γγ−1)=limt→0+N+1ρ+1(φ(t))−ρ[−(G(t)tg(t)−1γ−1)(tg′(t)g(t)+γ)+γ(G(t)tg(t)−1γ−1)−1γ−1(tg′(t)g(t)+γ)]=0, if (S1) holds and γ>(1+ρ)N+1N+1−ρ in (g2),0, if (S2) and (g3) hold with θ(N+1)ρ−N>γ>(1+ρ)N+1N+1−ρ.$$\begin{equation}\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{((N+1)G(t))^{N/(N+1)}}{g(t)\varphi(t)}-\frac{\gamma+N}{\gamma-1}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\frac{((N+1)G(t))^{N/(N+1)}}{g(t)}-\frac{\gamma+N}{\gamma-1}\varphi(t)}{(\varphi(t))^{\rho+1}}\\=&\lim\limits_{t\rightarrow0^{+}}\frac{\frac{-N((N+1)G(t))^{-1/(N+1)}g^{2}(t)-((N+1)G(t))^{N/(N+1)}g'(t)}{g^{2}(t)}-\frac{\gamma+N}{\gamma-1}\varphi'(t)}{(\rho+1)(\varphi(t))^{\rho}\varphi'(t)}\\=&\lim\limits_{t\rightarrow0^{+}}-\frac{N+1}{\rho+1}(\varphi(t))^{-\rho}\bigg(\frac{G(t)}{tg(t)}\frac{tg'(t)}{g(t)}+\frac{\gamma}{\gamma-1}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}\frac{N+1}{\rho+1}(\varphi(t))^{-\rho}\bigg[-\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)+\gamma\bigg(\frac{G(t)}{tg(t)}-\frac{1}{\gamma-1}\bigg)\\&-\frac{1}{\gamma-1}\bigg(\frac{tg'(t)}{g(t)}+\gamma\bigg)\bigg]=\begin{cases} 0,\,&\mbox{ if }\mathbf{(S_{1}) } \mbox{ holds and }\gamma>\frac{(1+\rho)N+1}{N+1-\rho}\mbox{ in }\mathbf{(g_{2})},\\0,\,&\mbox{ if }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{hold with}\frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho}.\end{cases}\end{split}\end{equation}$$(3.14)-(3.15) imply that (iii) holds.(iv) When (S1) holds in (g2), we haveg(ξt)ξNg(t)=ξ−(γ+N),t∈(0,a1].$$\frac{g(\xi t)}{\xi^{N}g(t)}=\xi^{-(\gamma+N)},\,t\in(0, a_{1}].$$In this case, for υ(t) = (−ln t)β or υ(t) = (φ(t))−ρ, we obtain(3.16)limt→0+υ(t)(g(ξt)ξNg(t)−ξ−(γ+N))=0.$$\begin{equation}\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{g(\xit)}{\xi^{N}g(t)}-\xi^{-(\gamma+N)}\bigg)=0.\end{equation}$$Now, we investigate the case that (S2) holds in (g2). If ξ = 1, then the result is obvious. If ξ ≠ 1, then we haveg(ξt)ξNg(t)−ξ−(γ+N)=ξ−(γ+N)(exp(∫ξttf(s)sds)−1).$$\frac{g(\xi t)}{\xi^{N}g(t)}-\xi^{-(\gamma+N)}=\xi^{-(\gamma+N)}\bigg(\exp\bigg(\int\limits_{\xit}^{t}\frac{f(s)}{s}ds\bigg)-1\bigg).$$It follows by Proposition 2.2 thatlimt→0+f(st)s=0andlimt→0+f(st)sf(t)=sθ−1$$\lim\limits_{t\rightarrow0^{+}}\frac{f(st)}{s}=0\, \mbox{and}\lim\limits_{t\rightarrow0^{+}}\frac{f(st)}{sf(t)}=s^{\theta-1}$$uniformly with respect to s ∈ [1, ξ] or s ∈ [ξ, 1]. So, we havelim t → 0 + ∫ ξ t t f ( τ ) τ d τ τ := t s _ _ lim t → 0 + ∫ ξ 1 f ( t s ) s d s = 0 $$ \lim\limits_{t\rightarrow0^{+}}\int\limits_{\xi t}^{t}\frac{f(\tau)}{\tau}d\tau^\underline{\underline{\tau:=ts}}\lim_{t\rightarrow0^{+}}\int\limits_{\xi}^{1}\frac{f(ts)}{s}ds=0 $$andlimt→0+∫ξ1f(st)sf(t)ds=∫ξ1sθ−1ds=−lnξ, if g3 holds with θ=0,1−ξθθ, if g3 holds with θ>0.$$\lim\limits_{t \rightarrow 0^{+}} \int\limits_{\xi}^{1} \frac{f(s t)}{s f(t)} d s=\int\limits_{\xi}^{1} s^{\theta-1} d s=\left\{\begin{array}{ll}-\ln \xi, & \text { if }\left(\mathbf{g}_{3}\right) \text { holds with } \theta=0, \\ \frac{1-\xi^{\theta}}{\theta}, & \text { if }\left(\mathbf{g}_{3}\right) \text { holds with } \theta>0.\end{array}\right.$$Since er−1 ∼ r as r → 0+, we conclude by (3.6), (3.8)-(3.10) thatlimt→0+υ(t)(g(ξt)ξNg(t)−ξ−(γ+N))=limt→0+ξ−(γ+N)υ(t)f(t)∫ξ1f(st)sf(t)ds=−σlnξξγ+N, if (g3)-(g4) hold with θ=0 in (g3),υ(t)=(−lnt)β,0, if (g3) hold with θ>0,υ(t)=(−lnt)β,0, if (g3) hold with θ(N+1)>(γ+N)ρ,υ(t)=(φ(t))−ρ.$$\begin{split}&\lim\limits_{t\rightarrow0^{+}}\upsilon(t)\bigg(\frac{g(\xi t)}{\xi^{N}g(t)}-\xi^{-(\gamma+N)}\bigg)\\=&\lim\limits_{t\rightarrow0^{+}}\xi^{-(\gamma+N)}\upsilon(t)f(t)\int\limits_{\xi}^{1}\frac{f(st)}{sf(t)}ds\\=&\begin{cases}-\frac{\sigma\ln\xi}{\xi^{\gamma+N}}, &\mbox{ if }\mathbf{(g_{3})}\mbox{-}\mathbf{(g_{4})} \mbox{ hold with } \theta=0 \mbox{ in }\mathbf{(g_{3})},\, \upsilon(t)=(-\ln t)^{\beta},\\0, &\mbox{ if }\mathbf{(g_{3})} \mbox{ hold with }\theta>0,\, \upsilon(t)=(-\ln t)^{\beta},\\0, &\mbox{ if }\mathbf{(g_{3})} \mbox{ hold with}\theta(N+1)>(\gamma+N)\rho,\,\upsilon(t)=(\varphi(t))^{-\rho}.\end{cases}\end{split}$$This, combined with (3.16), shows that (iv) holds. □Lemma 3.5(Lemma 2.3 in [48]) Let L˜∈L$\tilde{L}\in\mathcal {L}$, thenlimt→0+L˜(t)∫tt0L˜(s)sds=0.$$\lim\limits_{t\rightarrow0^{+}}\frac{\tilde{L}(t)}{\int_{t}^{t_{0}}\frac{\tilde{L}(s)}{s}ds}=0.$$If further ∫0t0L˜(s)sds<∞$ \int_{0}^{t_{0}}\frac{\tilde{L}(s)}{s}ds<\infty $, thenlimt→0+L˜(t)∫0tL˜(s)sds=0.$$\lim\limits_{t\rightarrow0^{+}}\frac{\tilde{L}(t)}{\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds}=0.$$Lemma 3.6Suppose that g satisfies (g1)-(g2). In particular, if (S2) holds in (g2), we suppose that (g3) holds, and if θ = 0 in (g3), we further suppose that (g4) holds. ψ isuniquely determined by (1.4). Then,ψ′(s)=((N+1)G(ψ(t)))1N+1,ψ(t)>0,ψ(0)=0$ \,\psi'(s)=((N+1)G(\psi(t)))^{\frac{1}{N+1}},\, \,\psi(t)>0,\, \psi(0)=0$and−ψ′′(t)=((N+1)G(ψ(t)))1−NN+1g(ψ(t)),t∈(0,a1);$$-\psi''(t)=((N+1)G(\psi(t)))^{\frac{1-N}{N+1}}g(\psi(t)),\, t\in(0, a_{1});$$−(ψ′(t))N−1ψ′′(t)=g(ψ(t));$-(\psi'(t))^{N-1}\psi''(t)=g(\psi(t));$limt→0+tψ′(t)ψ(t)=N+1γ+N,i.e.,ψ∈NRVZN+1γ+N;$\lim\limits_{t\rightarrow0^{+}}\frac{t\psi'(t)}{\psi(t)}=\frac{N+1}{\gamma+N},\, i.e.,\, \psi\in NRVZ_{\frac{N+1}{\gamma+N}};$limt→0+ψ′(t)tψ′′(t)=−γ+Nγ−1,i.e.,ψ′∈NRVZ−γ−1γ+N;$\lim\limits_{t\rightarrow0^{+}}\frac{\psi'(t)}{t\psi''(t)}=-\frac{\gamma+N}{\gamma-1},\, i.e.,\, \psi'\in NRVZ_{-\frac{\gamma-1}{\gamma+N}};$if k∈Λ,thenlimt→0+lntlnψ(K(t))=(γ+N)DkN+1;$k\in\Lambda, \,then\,\lim\limits_{t\rightarrow0^{+}}\frac{\ln t}{\ln\psi(K(t))}=\frac{(\gamma+N)D_{k}}{N+1};$if L˜∈Land(1.3)holds,thenlimt→0+lnψ((∫0tL˜(s)sds)NN+1)lnt=0$\tilde{L}\in\mathcal {L}\,and\, (1.3)\, holds,\, then\,\lim\limits_{t\rightarrow0^{+}}\frac{\ln\psi\big(\big(\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds\big)^{\frac{N}{N+1}}\big)}{\lnt}=0$;limt→0+(−lnψ(t))β(ψ′(t)tψ′′(t)+γ+Nγ−1)=(N+1)σ(γ−1)2,if(S2)and(g3)−(g4)holdwithθ=0in(g3),0,if(S1)holds(or(S2)and(g3)holdwithθ>0);$$\begin{split}&\lim\limits_{t\rightarrow0^{+}}(-\ln\psi(t))^{\beta}\bigg(\frac{\psi'(t)}{t\psi''(t)}+\frac{\gamma+N}{\gamma-1}\bigg)\\&=\begin{cases}\frac{(N+1)\sigma}{(\gamma-1)^{2}}, &\,{ if }\, \mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})}\,{-}\mathbf{(g_{4})} \,{ hold\, with }\,\theta=0 \,{ in }\,\mathbf{(g_{3})},\\0, &\,{ if }\,\mathbf{(S_{1})} \,{ holds\, (or }\,\mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})} \,{ hold \,with }\, \theta>0);\end{cases}\end{split}$$let ρ ∈ (0, 1], thenlim t → 0 + t − ρ ( ψ ′ ( t ) t ψ ″ ( t ) + γ + N γ − 1 ) = 0 , i f ( S 1 ) h o l d s a n d γ > ( 1 + ρ ) N + 1 N + 1 − ρ i n ( g 2 ) , 0 , i f ( S 2 ) a n d ( g 3 ) h o l d w i t h θ ( N + 1 ) ρ − N > γ > ( 1 + ρ ) N + 1 N + 1 − ρ ; $$\begin{split}&\lim_{t\rightarrow0^{+}}t^{-\rho}\bigg(\frac{\psi'(t)}{t\psi''(t)}+\frac{\gamma+N}{\gamma-1}\bigg)\\&=\begin{cases}0, &\,{ if }\,\mathbf{(S_{1})}\,{ holds\, and }\,\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \,{ in }\,\mathbf{(g_{2})},\\0, &\,{ if }\,\mathbf{(S_{2})} \,{ and }\,\mathbf{(g_{3})} \,{hold\, with }\,\frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho};\end{cases}\end{split}$$let ξ be a positive constant, thenlimt→0+(−lnψ(t))β(g(ξψ(t))ξNg(ψ(t))−ξ−(γ+N))=−σξ−(γ+N)lnξ,if(S2)and(g3)−(g4)holdwithθ=0in(g3),0,if(S1)holds(or(S2)and(g3)holdwithθ>0);$$\begin{split}&\lim\limits_{t\rightarrow0^{+}}(-\ln \psi(t))^{\beta}\bigg(\frac{g(\xi\psi(t))}{\xi^{N}g(\psi(t))}-\xi^{-(\gamma+N)}\bigg)\\&=\begin{cases}-\sigma\xi^{-(\gamma+N)}\ln\xi, &\,{ if }\,\mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})}\,{-}\mathbf{(g_{4})} \,{ hold\, with }\theta=0 \,{ in }\,\mathbf{(g_{3})},\\0, &\,{ if }\,\mathbf{(S_{1})} \,{ holds\, (or }\,\mathbf{(S_{2})}\,{ and }\,\mathbf{(g_{3})} \,{ hold \,with }\, \theta>0);\end{cases}\end{split}$$let (S1) hold (or (S2) and (g3) hold with θ(N + 1)>(γ+N)ρ), ξ be a positive constant and ρ ∈ (0, 1], then limt→0+t−ρ(g(ξψ(t))ξNg(ψ(t))−ξ−(γ+N))=0.$ \lim\limits_{t\rightarrow0^{+}}t^{-\rho}\big(\frac{g(\xi \psi(t))}{\xi^{N}g(\psi(t))}-\xi^{-(\gamma+N)}\big)=0. $Proof. (i)-(ii) By the definition of ψ, we obtain that (i)-(ii) hold.(iii)-(iv) It follows by (i) and Lemma 3.3 (iv)-(v) thatlim t → 0 + t ψ ′ ( t ) ψ ( t ) = lim t → 0 + t ( ( N + 1 ) G ( ψ ( t ) ) ) 1 N + 1 ψ ( t ) z := ψ ( t ) _ _ lim z → 0 + φ ( z ) ( ( N + 1 ) G ( z ) ) 1 N + 1 z = N + 1 γ + N $$ \lim\limits_{t\rightarrow0^{+}}\frac{t\psi'(t)}{\psi(t)}=\lim\limits_{t\rightarrow0^{+}}\frac{t((N+1)G(\psi(t)))^{\frac{1}{N+1}}}{\psi(t)}\underline{\underline{z:=\psi(t)}}\lim\limits_{z\rightarrow0^{+}}\frac{\varphi(z)((N+1)G(z))^{\frac{1}{N+1}}}{z}=\frac{N+1}{\gamma+N} $$andlimt→0+ψ′(t)tψ′′(t)=−limz→0+((N+1)G(z))NN+1g(z)φ(z)=−γ+Nγ−1.$$\lim\limits_{t\rightarrow0^{+}}\frac{\psi'(t)}{t\psi''(t)}=-\lim\limits_{z\rightarrow0^{+}}\frac{((N+1)G(z))^{\frac{N}{N+1}}}{g(z)\varphi(z)}=-\frac{\gamma+N}{\gamma-1}.$$(v) We conclude by (iii), Lemma 3.1 (ii) and Lemma 2.5 (iii) that (v) holds.(vi) By using Lemma 3.5, we havelimt→0+t((∫0tL˜(s)sds)NN+1)′(∫0tL˜(s)sds)NN+1=NN+1limt→0+L˜(t)∫0tL˜(s)sds=0.$$\lim\limits_{t\rightarrow0^{+}}\frac{t\big(\big(\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds\big)^{\frac{N}{N+1}}\big)'}{\big(\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds\big)^{\frac{N}{N+1}}}=\frac{N}{N+1}\lim\limits_{t\rightarrow0^{+}}\frac{\tilde{L}(t)}{\int_{0}^{t}\frac{\tilde{L}(s)}{s}ds}=0.$$This implies that(3.17)t↦(∫0tL˜(s)sds)NN+1∈NRVZ0.$$\begin{equation}t\mapsto\bigg(\int\limits_{0}^{t}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\inNRVZ_{0}.\end{equation}$$So, by (3.17), (iii) and Proposition 2.7 we further obtain(3.18)t↦ψ((∫0tL˜(s)sds)NN+1)∈NRVZ0.$$\begin{equation}t\mapsto\psi\bigg(\bigg(\int\limits_{0}^{t}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\inNRVZ_{0}.\end{equation}$$We have by Proposition 2.5 (iii) that (vi) holds.(vii)-(viii) It follows by (i) and Lemma 3.4 (iii) thatlim t → 0 + ( − ln ⁡ ψ ( t ) ) β ( ψ ′ ( t ) t ψ ″ ( t ) + γ + N γ − 1 ) = lim t → 0 + ( − ln ⁡ ψ ( t ) ) β ( − ( ( N + 1 ) G ( ψ ( t ) ) ) N N + 1 g ( ψ ( t ) ) t + γ + N γ − 1 ) z := ψ ( t ) _ _ lim z → 0 + ( − ln ⁡ z ) β ( − ( ( N + 1 ) G ( z ) ) N N + 1 g ( z ) φ ( z ) + γ + N γ − 1 ) = ( N + 1 ) σ ( γ − 1 ) 2 , if ( S 2 ) and ( g 3 ) - ( g 4 ) hold with θ = 0 in ( g 3 ) , 0 , if ( S 1 ) holds (or ( S 2 ) and ( g 3 ) hold with θ > 0 ) , $$ \begin{split} &\lim\limits_{t\rightarrow0^{+}}(-\ln \psi(t))^{\beta}\bigg(\frac{\psi'(t)}{t\psi''(t)}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &=\lim\limits_{t\rightarrow0^{+}}(-\ln \psi(t))^{\beta}\bigg(-\frac{((N+1)G(\psi(t)))^{\frac{N}{N+1}}}{g(\psi(t))t}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &\underline{\underline{z:=\psi(t)}}\lim\limits_{z\rightarrow0^{+}}(-\ln z)^{\beta}\bigg(-\frac{((N+1)G(z))^{\frac{N}{N+1}}}{g(z)\varphi(z)}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &=\begin{cases} \frac{(N+1)\sigma}{(\gamma-1)^{2}}, &\mbox{ if }\mathbf{(S_{2})}\mbox{ and }\mathbf{(g_{3})}\mbox{-}\mathbf{(g_{4})} \mbox{ hold with }\theta=0 \mbox{ in }\mathbf{(g_{3})},\\ 0, &\mbox{ if } \mathbf{(S_{1})} \mbox{ holds (or }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with } \theta>0), \end{cases} \end{split} $$andlim t → 0 + t − ρ ( ψ ′ ( t ) t ψ ″ ( t ) + γ + N γ − 1 ) = lim t → 0 + t − ρ ( − ( ( N + 1 ) G ( ψ ( t ) ) ) N N + 1 g ( ψ ( t ) ) t + γ + N γ − 1 ) z := ψ ( t ) _ _ lim z → 0 + ( φ ( z ) ) − ρ ( − ( ( N + 1 ) G ( z ) ) N N + 1 g ( z ) φ ( z ) + γ + N γ − 1 ) = 0 , if ( S 1 ) holds and γ > ( 1 + ρ ) N + 1 N + 1 − ρ in ( g 2 ) , 0 , if ( S 2 ) and ( g 3 ) hold with θ ( N + 1 ) ρ − N > γ > ( 1 + ρ ) N + 1 N + 1 − ρ . $$ \begin{split} \lim\limits_{t\rightarrow0^{+}}t^{-\rho}\bigg(\frac{\psi'(t)}{t\psi''(t)}+\frac{\gamma+N}{\gamma-1}\bigg) &=\lim\limits_{t\rightarrow0^{+}}t^{-\rho}\bigg(-\frac{((N+1)G(\psi(t)))^{\frac{N}{N+1}}}{g(\psi(t))t}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &\underline{\underline{z:=\psi(t)}}\lim\limits_{z\rightarrow0^{+}}(\varphi(z))^{-\rho}\bigg(-\frac{((N+1)G(z))^{\frac{N}{N+1}}}{g(z)\varphi(z)}+\frac{\gamma+N}{\gamma-1}\bigg)\\ &=\begin{cases} 0, &\mbox{ if }\mathbf{(S_{1})}\mbox{ holds and }\gamma>\frac{(1+\rho)N+1}{N+1-\rho} \mbox{ in }\mathbf{(g_{2})},\\ 0, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with } \frac{\theta(N+1)}{\rho}-N>\gamma>\frac{(1+\rho)N+1}{N+1-\rho}. \end{cases} \end{split} $$(ix)-(x) It follows by Lemma 3.4 (iv) thatlim t → 0 + ( − ln ⁡ ψ ( t ) ) β ( g ( ξ ψ ( t ) ) ξ N g ( ψ ( t ) ) − ξ − ( γ + N ) ) z := ψ ( t ) _ _ lim z → 0 + ( − ln ⁡ z ) β ( g ( ξ z ) ξ N g ( z ) − ξ − ( γ + N ) ) = − σ ξ − ( γ + N ) ln ⁡ ξ , if ( S 2 ) and ( g 3 ) - ( g 4 ) hold with θ = 0 in ( g 3 ) , 0 , if ( S 1 ) holds (or ( S 2 ) and ( g 3 ) hold with θ > 0 ) . $$ \begin{split} &\lim\limits_{t\rightarrow0^{+}}(-\ln \psi(t))^{\beta}\bigg(\frac{g(\xi \psi(t))}{\xi^{N} g(\psi(t))}-\xi^{-(\gamma+N)}\bigg)\\ &\underline{\underline{z:=\psi(t)}}\lim\limits_{z\rightarrow0^{+}}(-\ln z)^{\beta}\bigg(\frac{g(\xi z)}{\xi^{N} g(z)}-\xi^{-(\gamma+N)}\bigg)\\ &=\begin{cases} -\sigma\xi^{-(\gamma+N)}\ln\xi, &\mbox{ if }\mathbf{(S_{2})}\mbox{ and }\mathbf{(g_{3})}\mbox{-}\mathbf{(g_{4})} \mbox{ hold with }\theta=0 \mbox{ in }\mathbf{(g_{3})},\\ 0, &\mbox{ if }\mathbf{(S_{1})} \mbox{ holds (or }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with } \theta>0). \end{cases} \end{split}$$Moreover, if (S1) holds or (S2) and (g3) hold with θ(N + 1)>(γ+N)ρ, then by Lemma 3.4 (iv), we havelimt→0+t−ρg(ξψ(t))ξNg(ψ(t))−ξ−(γ+N)=z:=ψ(t)limz→0+(φ(z))−ρg(ξz)ξNg(z)−ξ−(γ+N)=0.$$\lim\limits_{t \rightarrow 0^{+}} t^{-\rho}\left(\frac{g(\xi \psi(t))}{\xi^{N} g(\psi(t))}-\xi^{-(\gamma+N)}\right) \stackrel{z:=\psi(t)}{=} \lim\limits_{z \rightarrow 0^{+}}(\varphi(z))^{-\rho}\left(\frac{g(\xi z)}{\xi^{N} g(z)}-\xi^{-(\gamma+N)}\right)=0.$$□4The Second Boundary BehaviorIn this section, we prove Theorems 1.1-1.3. We first introduce some lemmas as follows.Lemma 4.1(Lemma 2.1 in [27]) Let Ω be a bounded domain in ℝN with N ≥ 2, and let u,v∈C(Ωˉ)∩C2(Ω)$ u,\,v\in C(\bar{\Omega})\cap C^{2}(\Omega) $. Suppose h(x, t) is defined for x ∈ Ω and t in some interval containingthe ranges of u and v. If the following hold:(i) h is strictly increasing in t for all x ∈ Ω,(ii) the matrix D2v is positive definite in Ω,(iii) det(D2v)=h(x, v) and det (D2u) ⩽ h(x, u), x ∈ Ω,(iv) u=v on ∂Ω,then, we have u≥v in Ω.For any δ>0, letΩδ={x∈Ω:0<d(x)<δ}.$$\Omega_{\delta}=\{x\in\Omega: 0<d(x)<\delta\}.$$Since Ω is Cm-smooth for m≥2, we can always takeδ1 > 0 such that (see Lemmas 14.16 and 14.17 in[18])d∈Cm(Ωδ1)and|∇d(x)|=1,x∈Ωδ1.$$d\in C^{m}(\Omega_{\delta_{1}}) \mbox{ and } |\nablad(x)|=1,\,x\in\Omega_{\delta_{1}}.$$Let xˉ∈∂Ω$ \bar{x}\in\partial\Omega $be the projection of the pointx∈Ωδ1$ x\in\Omega_{\delta_{1}} $to ∂Ω, andκi(xˉ)$ \kappa_{i}(\bar{x}) $(i=1, ···, N − 1) be the principalcurvatures of ∂Ω at xˉ$ \bar{x} $, thenD2(d(x))=diag(−κ1(xˉ)1−d(x)κ1(xˉ),⋅⋅⋅,−κN−1(xˉ)1−d(x)κN−1(xˉ),0).$$D^{2}(d(x))=\mbox{diag}\bigg(\frac{-\kappa_{1}(\bar{x})}{1-d(x)\kappa_{1}(\bar{x})},\cdot\cdot\cdot,\frac{-\kappa_{N-1}(\bar{x})}{1-d(x)\kappa_{N-1}(\bar{x})},0\bigg).$$Lemma 4.2(See the proof of Proposition 2.4 in [27], Proposition 2.1 and Corollary 2.3 in [9]) Let h be a C2-function on (0, δ1), thendet(D2h(d(x)))=(−h′(d(x)))N−1h′′(d(x))∏i=1N−1κi(xˉ)1−d(x)κi(xˉ),x∈Ωδ1.$$\mbox{det}(D^{2}h(d(x)))=(-h'(d(x)))^{N-1}h''(d(x))\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})},\,x\in\Omega_{\delta_{1}}.$$Lemma 4.3(Corollary 2.3 in [20]) Let h be a C2-function on (0, δ1) and Ω be a bounded domain with ∂Ω ∈ Cm for m=2, then, for i=1, ···, N,Si(D2(h(d(x))))=(−h′(d(x)))iSi(ϵ1,⋅⋅⋅,ϵN−1)+(−h′(d(x)))i−1h′′(d(x))Si−1(ϵ1,⋅⋅⋅,ϵN−1),x∈Ωδ1$$S_{i}(D^{2}(h(d(x))))=(-h'(d(x)))^{i}S_{i}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1})+(-h'(d(x)))^{i-1}h''(d(x))S_{i-1}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1}),\,x\in\Omega_{\delta_{1}}$$whereϵj=κj(xˉ)1−κj(xˉ)d(x),j=1,⋅⋅⋅,N−1.$$\epsilon_{j}=\frac{\kappa_{j}(\bar{x})}{1-\kappa_{j}(\bar{x})d(x)},\,j=1,\cdot\cdot\cdot,N-1.$$Definition 4.4(Definition 1.1 in [42]) Let i ∈ {1, ···, N} and Ω be an open bounded subset of ℝN; a function u ∈ C2(Ω) is (strictly) i-convex if Sl(D2u)(>)⩾0 in Ω for l=1, ···, i. In particular, if i=N, then we say that u is (strictly) convex in Ω.4.1Proof of Theorem 1.1Next, we prove Theorem 1.1 and we first show some preliminaries as follows.Fix ε>0 and letw±(d(x))=ξ±ψ(K(d(x)))(1+(C±±ε)(−lnd(x))−β),x∈Ωδ1,$$w_{\pm}(d(x))=\xi_{\pm}\psi(K(d(x)))\big(1+(C_{\pm}\pm\varepsilon)(-\lnd(x))^{-\beta}\big),\,x\in\Omega_{\delta_{1}},$$where ξ ± and C ± are in Theorem 1.1. By the Lagrange’s mean value theorem, we obtain that there exist λ ± ∈ (0, 1) and(4.1)Θ±(d(x))=ξ±ψ(K(d(x)))(1+λ±(C±±ε)(−lnd(x))−β)$$\begin{equation}\Theta_{\pm}(d(x))=\xi_{\pm}\psi(K(d(x)))\big(1+\lambda_{\pm}(C_{\pm}\pm\varepsilon)(-\lnd(x))^{-\beta}\big)\end{equation}$$such that for any x∈Ωδ1$ x\in\Omega_{\delta_{1}} $g(w±(d(x)))=g(ξ±ψ(K(d(x))))+ξ±ψ(K(d(x)))g′(Θ±(d(x)))(C±±ε)(−lnd(x))−β.$$\begin{split}g(w_{\pm}(d(x)))&=g(\xi_{\pm}\psi(K(d(x))))+\xi_{\pm}\psi(K(d(x)))g'(\Theta_{\pm}(d(x)))(C_{\pm}\pm\varepsilon)(-\lnd(x) )^{-\beta}.\end{split}$$Since g ∈ NRVZ−γ, we have by Proposition 2.2 that(4.2)limd(x)→0g(ξ±ψ(K(d(x))))g(Θ±(d(x)))=limd(x)→0g′(ξ±ψ(K(d(x))))g′(Θ±(d(x)))=1.$$\begin{equation}\lim\limits_{d(x)\rightarrow0}\frac{g(\xi_{\pm}\psi(K(d(x))))}{g(\Theta_{\pm}(d(x)))}=\lim\limits_{d(x)\rightarrow0}\frac{g'(\xi_{\pm}\psi\big(K(d(x))))}{g'(\Theta_{\pm}(d(x)))}=1.\end{equation}$$Moreover, by the definitions of ξ ± and C ± , we can take a sufficiently small positive constant still denoted by δ1 such that(4.3)w+(d(x))≥w−(d(x)),x∈Ωδ1.$$\begin{equation}w_{+}(d(x))\geq w_{-}(d(x)),\,x\in\Omega_{\delta_{1}}.\end{equation}$$Proof. Our proof is divided into two steps and the outline of the proof is given as below.• In Step 1, for fixed ε>0 andx∈Ωδ1$ x\in\Omega_{\delta_{1}} $, we first give some functions I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) (which are corresponding to ε>0), and then by detailed calculation we will show that there exists a sufficiently small positive constant δε<δ1 such that(4.4)I1+(d(x))+I2+(d(x))+I3+(d(x))>0,x∈Ωδε$$\begin{equation}I_{1+}(d(x))+I_{2+}(d(x))+I_{3+}(d(x))>0,\,x\in\Omega_{\delta_{\varepsilon}}\end{equation}$$and(4.5)I1−(d(x))+I2−(d(x))+I3−(d(x))<0,x∈Ωδε.$$\begin{equation}I_{1-}(d(x))+I_{2-}(d(x))+I_{3-}(d(x))<0,\,x\in\Omega_{\delta_{\varepsilon}}.\end{equation}$$• In Step 2, we will define two functions u_ε,u‾ε$ \underline{u}_{\varepsilon}, \overline{u}_{\varepsilon} $inΩδε$ \Omega_{\delta_{\varepsilon}} $and show they are sub-and super-solutions of Eq. (1.1) in Ωδε$ \Omega_{\delta_{\varepsilon}} $by (4.4) and (4.5), respectively. In particular, we will show u_ε$ \underline{u}_{\varepsilon} $is strictly convex inΩδε$ \Omega_{\delta_{\varepsilon}} $. Finally, we will establish the second boundary behavior of the unique strictly convex solution to problem (1.1) by using Lemma 4.1.By the above analysis, we see that how to get (4.4)-(4.5) is the key of the research.Step 1.We first define functions I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as follows.For fixed ε>0 and ∀x∈Ωδ1$ \forall\,x\in\Omega_{\delta_{1}} $, we defineI1±(d(x))=(−lnd(x))β[(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ±−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))];I2±(d(x))=C±[N(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)A±∏i=1N−1(1−d(x)κi(xˉ))−1−ξ±ψ(K(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))g′(Θ±(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))]±ε[N(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ+∏i=1N−1(1−d(x)κi(xˉ))−1−ξ±ψ(K(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))g′(Θ±(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))];I3±(d(x))=(−lnd(x))β(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)×mˆ±∑i=1N−1CN−1i(−1)i+1(m±d(x))i(1−d(x)m±)N−1+ζ±(d(x))(−lnd(x))−β∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)−(B0±ε)g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))(d(x))μ(−lnd(x))β−(B0±ε)(C±±ε)ξ±ψ(K(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))g′(Θ±(d(x)))g′(ξ±ψ(K(d(x))))g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x))))(d(x))μ,$$\begin{split}&I_{1\pm}(d(x))=(-\lnd(x))^{\beta}\bigg[\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\hat{m}_{\pm}\\&-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg];\\&I_{2\pm}(d(x))=C_{\pm}\bigg[N\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\mathscr{A}_{\pm}\prod\limits_{i=1}^{N-1}(1-d(x)\kappa_{i}(\bar{x}))^{-1}\\&-\frac{\xi_{\pm}\psi(K(d(x)))g'(\xi_{\pm}\psi(K(d(x))))}{g(\xi_{\pm}\psi(K(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\xi_{\pm}\psi(K(d(x))))}\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg]\\&\pm\varepsilon\bigg[N\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\hat{m}_{+}\prod\limits_{i=1}^{N-1}(1-d(x)\kappa_{i}(\bar{x}))^{-1}\\&-\frac{\xi_{\pm}\psi(K(d(x)))g'(\xi_{\pm}\psi(K(d(x))))}{g(\xi_{\pm}\psi(K(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\xi_{\pm}\psi(K(d(x))))}\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg];\\&I_{3\pm}(d(x))=(-\lnd(x))^{\beta}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\\&\times\frac{\hat{m}_{\pm}\sum_{i=1}^{N-1}C_{N-1}^{i}(-1)^{i+1}(m_{\pm}d(x))^{i}}{(1-d(x)m_{\pm})^{N-1}}+\frac{\zeta_{\pm}(d(x))}{(-\lnd(x))^{-\beta}}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\\&-(B_{0}\pm\varepsilon)\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}(d(x))^{\mu}(-\lnd(x))^{\beta}\\&-(B_{0}\pm\varepsilon)(C_{\pm}\pm\varepsilon)\frac{\xi_{\pm}\psi(K(d(x)))g'(\xi_{\pm}\psi(K(d(x))))}{g(\xi_{\pm}\psi(K(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\xi_{\pm}\psi(K(d(x))))}\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}(d(x))^{\mu},\end{split}$$where(S2) holds and θ=0k∈Λ1A±=mˆ±, if A+≤0 and A−≤0,A±=mˆ∓, if A+≥0 and A−≥0,A+=A−=mˆ−, if A+>0 and A−<0,A+=A−=mˆ+, if A+<0 and A−>0;k∈Λ2,βA±=mˆ±, if A++B≤0 and A++B≤0,A±=mˆ∓, if A++B≥0 and A++B≥0,A+=A−=mˆ−, if A++B>0 and A++B<0,A+=A−=mˆ+, if A++B<0 and A++B>0;(S1) holdsk∈Λ1,A±=mˆ±;k∈Λ2,β,A+=mˆ−, if B≥0,A−=mˆ+, if B<0;(S2) holds and θ>0k∈Λ2,β,A+=mˆ−, if B≥0,A−=mˆ+, if B<0,$$\begin{cases}\mathbf{(S_{2})}\ \mbox{ holds and }\theta=0&\begin{cases}k\in\Lambda_{1}&\begin{cases}\mathscr{A}_{\pm}=\hat{m}_{\pm},\, &\mbox{ if }A_{+}\leq0 \mbox{ and }A_{-}\leq0,\\\mathscr{A}_{\pm}=\hat{m}_{\mp},\,&\mbox{if }A_{+}\geq0 \mbox{ and }A_{-}\geq0,\\\mathscr{A}_{+}=\mathscr{A}_{-}=\hat{m}_{-},\,&\mbox{if }A_{+}>0 \mbox{ and }A_{-}<0,\\\mathscr{A}_{+}=\mathscr{A}_{-}=\hat{m}_{+},\,&\mbox{if }A_{+}<0 \mbox{ and }A_{-}>0;\\\end{cases}\\k\in\Lambda_{2,\beta}&\begin{cases}\mathscr{A}_{\pm}=\hat{m}_{\pm},\, &\mbox{ if }\mathfrak{A}_{+}+\mathfrak{B}\leq0 \mbox{ and }\mathfrak{A}_{+}+\mathfrak{B}\leq0,\\\mathscr{A}_{\pm}=\hat{m}_{\mp},\,&\mbox{if }\mathfrak{A}_{+}+\mathfrak{B}\geq0 \mbox{ and }\mathfrak{A}_{+}+\mathfrak{B}\geq0,\\\mathscr{A}_{+}=\mathscr{A}_{-}=\hat{m}_{-},\,&\mbox{if }\mathfrak{A}_{+}+\mathfrak{B}>0 \mbox{ and }\mathfrak{A}_{+}+\mathfrak{B}<0,\\\mathscr{A}_{+}=\mathscr{A}_{-}=\hat{m}_{+},\,&\mbox{if }\mathfrak{A}_{+}+\mathfrak{B}<0 \mbox{ and }\mathfrak{A}_{+}+\mathfrak{B}>0;\\\end{cases}\end{cases}\\\mathbf{(S_{1})} \mbox{ holds}&\begin{cases}k\in\Lambda_{1}, &\mathscr{A}_{\pm}=\hat{m}_{\pm};\\k\in\Lambda_{2,\beta}, &\begin{cases}\mathscr{A}_{+}=\hat{m}_{-},\,&\mbox{ if }\mathfrak{B}\geq0,\\\mathscr{A}_{-}=\hat{m}_{+},\,&\mbox{ if }\mathfrak{B}<0;\end{cases}\end{cases}\\\mathbf{(S_{2})}\ \mbox{ holds and }\theta>0&\begin{cases}k\in\Lambda_{2,\beta}, &\begin{cases}\mathscr{A}_{+}=\hat{m}_{-},\,&\mbox{ if }\mathfrak{B}\geq0,\\\mathscr{A}_{-}=\hat{m}_{+},\,&\mbox{ if }\mathfrak{B}<0,\end{cases}\end{cases}\end{cases}$$(4.6)m+:=min{minxˉ∈∂Ωκi(xˉ):i=1,⋅⋅⋅,N−1} and m−:=max{maxxˉ∈∂Ωκi(xˉ):i=1,⋅⋅⋅,N−1}$$\begin{equation}m_{+}:=\min\{\min\limits_{\bar{x}\in\partial\Omega}\kappa_{i}(\bar{x}):i=1,\cdot\cdot\cdot,N-1\}\mbox{ and }m_{-}:=\max\{\max\limits_{\bar{x}\in\partial\Omega}\kappa_{i}(\bar{x}):i=1,\cdot\cdot\cdot,N-1\}\end{equation}$$andζ±(d(x))=(C±±ε)2(N−1)(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(−lnd(x))−2β+2β(C±±ε)ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)(−lnd(x))−β−1×(1+(C±±ε)(N−1)(−lnd(x))−β+ν±(d(x)))−β(C±±ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2(−lnd(x))−β−1×(1−(β+1)(−lnd(x))−1)(1+(C±±ε)(N−1)(−lnd(x))−β+ν±(d(x)))+ν±(d(x))(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(1+(C±±ε)(−lnd(x))−β),$$\begin{align*}\zeta_{\pm}(d(x))&=(C_{\pm}\pm\varepsilon)^{2}(N-1)\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)(-\lnd(x))^{-2\beta}\\&+2\beta(C_{\pm}\pm\varepsilon)\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{k(d(x))d(x)}(-\lnd(x))^{-\beta-1}\\&\times\big(1+(C_{\pm}\pm\varepsilon)(N-1)(-\lnd(x))^{-\beta}+\nu_{\pm}(d(x))\big)\\&-\beta(C_{\pm}\pm\varepsilon)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}(-\lnd(x))^{-\beta-1}\\&\times\big(1-(\beta+1)(-\lnd(x))^{-1}\big)\big(1+(C_{\pm}\pm\varepsilon)(N-1)(-\lnd(x))^{-\beta}+\nu_{\pm}(d(x))\big)\\&+\nu_{\pm}(d(x))\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\big(1+(C_{\pm}\pm\varepsilon)(-\lnd(x))^{-\beta}\big),\end{align*}$$where A ± are given by (1.8), A±$ \mathfrak{A}_{\pm} $and B$ \mathfrak{B} $are given by (1.11) andν±(d(x))=R2±(d(x))+∑i=2N−1CN−1i(R1±(d(x))+R2±(d(x)))i,$$\nu_{\pm}(d(x))=\mathfrak{R}_{2\pm}(d(x))+\sum_{i=2}^{N-1}C_{N-1}^{i}\big(\mathfrak{R}_{1\pm}(d(x))+\mathfrak{R}_{2\pm}(d(x))\big)^{i},$$where(4.7)CN−1i=(N−1)!i!(N−1−i)!,R1±(d(x))=(C±±ε)(−lnd(x))−β,$$\begin{equation}C_{N-1}^{i}=\frac{(N-1)!}{i!(N-1-i)!},\,\,\,\mathfrak{R}_{1\pm}(d(x))=(C_{\pm}\pm\varepsilon)(-\lnd(x))^{-\beta},\end{equation}$$andR2±(d(x))=(N−1)β(C±±ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)(−lnd(x))−β−1.$$\mathfrak{R}_{2\pm}(d(x))=(N-1)\beta(C_{\pm}\pm\varepsilon)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{k(d(x))d(x)}(-\lnd(x))^{-\beta-1}.$$Next, we prove(4.8)limd(x)→0(I1±(d(x))+I2±(d(x))+I3±(d(x)))=±ε(γ+Nmˆ+mˆ±)ξ±−(γ+N).$$\begin{equation}\lim_{d(x)\rightarrow0}(I_{1\pm}(d(x))+I_{2\pm}(d(x))+I_{3\pm}(d(x)))=\pm\varepsilon\bigg(\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{\pm}}\bigg)\xi_{\pm}^{-(\gamma+N)}.\end{equation}$$To prove (4.8), we calculate the limits of I1 ± (d(x)),I2 ± (d(x)) and I3 ± (d(x)) as d(x) → 0.• First, by Lemma 3.1 (ii), Lemma 3.6 (v), (vii) and (ix), we obtain that(4.9)limd(x)→0I1±(d(x))=ϖ1±+ϖ2±, if (S2) and (g3) hold with θ=0,k∈Λ1,ϖ1±+ϖ2±+(γ+N)E2,kmˆ±γ−1, if (S2) and (g3) hold with θ=0,k∈Λ2,β,γ+Nγ−1E2,kmˆ±, if (S2) and (g3) hold with θ>0,k∈Λ2,β,γ+Nγ−1E2,kmˆ±, if (S1) holds,k∈Λ2,β,0, if (S1) holds,k∈Λ1,$$\begin{equation}\lim\limits_{d(x)\rightarrow0}I_{1\pm}(d(x))=\begin{cases}\varpi_{1\pm}+\varpi_{2\pm}, &\mbox{ if } \mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with }\theta=0, k\in\Lambda_{1}, \\\varpi_{1\pm}+\varpi_{2\pm}+\frac{(\gamma+N)E_{2,k}\hat{m}_{\pm}}{\gamma-1}, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with }\theta=0,\, k\in\Lambda_{2, \beta},\\\frac{\gamma+N}{\gamma-1}E_{2,k}\hat{m}_{\pm}, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and }\mathbf{(g_{3})} \mbox{ hold with}\theta>0,k\in\Lambda_{2, \beta},\\\frac{\gamma+N}{\gamma-1}E_{2,k}\hat{m}_{\pm},&\mbox{ if }\mathbf{(S_{1})}\mbox{ holds}, k\in\Lambda_{2, \beta},\\0, &\mbox{ if }\mathbf{(S_{1})} \mbox{ holds}, k\in\Lambda_{1},\\\end{cases}\end{equation}$$whereϖ1±=((γ+N)DkN+1)β(N+1)(1−Dk)σmˆ±(γ−1)2andϖ2±=((γ+N)DkN+1)βσlnξ±ξ±γ+N.$$\varpi_{1\pm}=\bigg(\frac{(\gamma+N)D_{k}}{N+1}\bigg)^{\beta}\frac{(N+1)(1-D_{k})\sigma\hat{m}_{\pm}}{(\gamma-1)^{2}} \mbox{ and }\varpi_{2\pm}=\bigg(\frac{(\gamma+N)D_{k}}{N+1}\bigg)^{\beta}\frac{\sigma\ln\xi_{\pm}}{\xi_{\pm}^{\gamma+N}}.$$• Second, by (4.2), Lemma 3.1 (ii), Lemma 3.6 (iv) and thechoices of ξ ± , C ± in Theorem 1.1, we obtain(4.10)limd(x)→0I2±(d(x))=C±[(γ+N)Dk−(N+1)]Nγ−1A±+γξ±−(γ+N)±ε(γ+Nmˆ+mˆ±)ξ±−(γ+N).$$\begin{equation}\lim_{d(x)\rightarrow0}I_{2\pm}(d(x))=C_{\pm}\frac{[(\gamma+N)D_{k}-(N+1)]N}{\gamma-1}\mathscr{A}_{\pm}+\gamma\xi_{\pm}^{-(\gamma+N)}\pm\varepsilon\bigg(\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{\pm}}\bigg)\xi_{\pm}^{-(\gamma+N)}.\end{equation}$$• Third, by Lemma 3.1 (ii), Lemma 3.6 (iii)-(iv) and a straightforward calculation, we obtain(4.11)limd(x)→0I3±(d(x))=0.$$\begin{equation}\lim_{d(x)\rightarrow0}I_{3\pm}(d(x))=0.\end{equation}$$Combining (4.9) and (4.10)-(4.11), we obtain (4.8) holds. By (b1)-(b2) and (4.8), we see that there exists asufficiently small constant δε<δ1$ \delta_{\varepsilon}<\delta_{1} $such that(4.12)kN+1(d(x))1+B0−ε(d(x))μ≤b(x)≤kN+1(d(x))1+B0+ε(d(x))μ,x∈Ωδε$$\begin{equation}k^{N+1}(d(x))\left(1+\left(B_{0}-\varepsilon\right)(d(x))^{\mu}\right) \leq b(x) \leq k^{N+1}(d(x))\left(1+\left(B_{0}+\varepsilon\right)(d(x))^{\mu}\right), x \in \Omega_{\delta_{\varepsilon}}\end{equation}$$and (4.4)-(4.5) hold.Step 2. Letu_ε(x)=−ξ+ψ(K(d(x)))(1+(C++ε)(−lnd(x))−β),x∈Ωδε.$$\underline{u}_{\varepsilon}(x)=-\xi_{+}\psi(K(d(x)))\big(1+(C_{+}+\varepsilon)(-\lnd(x))^{-\beta}\big),\,x\in\Omega_{\delta_{\varepsilon}}.$$Theng(−u_ε(x))=g(ξ+ψ(K(d(x))))+ξ+ψ(K(d(x)))g′(Θ+(d(x)))(C++ε)(−lnd(x))−β,x∈Ωδε,$$\begin{split}g(-\underline{u}_{\varepsilon}(x))&=g(\xi_{+}\psi(K(d(x))))+\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(C_{+}+\varepsilon)(-\lnd(x) )^{-\beta},\,x\in\Omega_{\delta_{\varepsilon}},\end{split}$$where Θ+(d(x))$\Theta_{+}(d(x))$is given by (4.1). By Lemma 4.2, we have for any x∈Ωδε$x \in \Omega_{\delta_{\varepsilon}}$(4.13)det D 2 u _ ε ( x ) − b ( x ) g − u _ ε ( x ) ≥ − ξ + N ψ ′ ( K ( d ( x ) ) ) N − 1 ψ ′ ′ ( K ( d ( x ) ) ) k N + 1 ( d ( x ) ) × 1 + C + + ε ( N − 1 ) ( − ln ⁡ d ( x ) ) − β + v + ( d ( x ) ) × ψ ′ ( K ( x ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 1 + C + + ε ( − ln ⁡ d ( x ) ) − β + 2 β C + + ε × ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ( d ( x ) ) d ( x ) ( − ln ⁡ d ( x ) ) − β − 1 − β C + + ε ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 ( − ln ⁡ d ( x ) ) − β − 1 × 1 − ( β + 1 ) ( − ln ⁡ d ( x ) ) − 1 ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − k N + 1 ( d ( x ) ) 1 + B 0 + ε ( d ( x ) ) μ × g ξ + ψ ( K ( d ( x ) ) ) + ξ + ψ ( K ( d ( x ) ) ) g ′ Θ + ( d ( x ) ) C + + ε ( − ln ⁡ d ( x ) ) − β ≥ − ξ + N ψ ′ ( K ( d ( x ) ) ) N − 1 ψ ′ ′ ( K ( d ( x ) ) ) k N + 1 ( d ( x ) ) ( − ln ⁡ d ( x ) ) − β ( − ln ⁡ d ( x ) ) β l × ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 ∏ i = 1 N − 1 m ^ + 1 − d ( x ) m + N − 1 − m ^ + + m ^ + − g ( ξ + ψ ( K ( d ( x ) ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) + C + + ε N ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − ξ + ψ ( K ( d ( x ) ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ξ + ψ ( K ( d ( x ) ) ) × g ′ Θ + ( d ( x ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ξ + ψ ( K ( d ( x ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) + ζ + ( d ( x ) ) ( − ln ⁡ d ( x ) ) − β ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − B 0 + ε g ξ + ψ ( K ( d ( x ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) ( d ( x ) ) μ ( − ln ⁡ d ( x ) ) β − B 0 + ε C + + ε × ξ + ψ ( K ( d ( x ) ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ( ξ + ψ ( K ( d ( x ) ) ) ) g ′ Θ + ( d ( x ) ) g ′ ξ + ψ ( K ( d ( x ) ) ) g ξ + ψ ( K ( d ( x ) ) ) ξ + N g ( ψ ( K ( d ( x ) ) ) ) ( d ( x ) ) μ ≥ − ξ + N ψ ′ ( K ( d ( x ) ) ) N − 1 ψ ′ ′ ( K ( d ( x ) ) ) k N + 1 ( d ( x ) ) ( − ln ⁡ d ( x ) ) − β ∑ i = 1 3 I i + ( d ( x ) ) > 0 , $$\begin{equation}\begin{array}{l}\operatorname{det}\left(D^{2} \underline{u}_{\varepsilon}(x)\right)-b(x) g\left(-\underline{u}_{\varepsilon}(x)\right) \geq-\xi_{+}^{N}\left(\psi^{\prime}(K(d(x)))\right)^{N-1} \psi^{\prime \prime}(K(d(x))) k^{N+1}(d(x)) \\ \times\left(1+\left(C_{+}+\varepsilon\right)(N-1)(-\ln d(x))^{-\beta}+v_{+}(d(x))\right) \\ \times\left[\left(\frac{\left.\psi^{\prime}(K(x))\right)}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\left(1+\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}\right)+2 \beta\left(C_{+}+\varepsilon\right)\right. \\ \times \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{k(d(x)) d(x)}(-\ln d(x))^{-\beta-1} \\ -\beta\left(C_{+}+\varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\left.\psi^{\prime}(K(d(x)))\right)}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}(-\ln d(x))^{-\beta-1} \\ \left.\times\left(1-(\beta+1)(-\ln d(x))^{-1}\right)\right] \prod\limits_{i=1}^{N-1} \frac{\kappa_{i}(\bar{x})}{1-d(x) \kappa_{i}(\bar{x})}-k^{N+1}(d(x))\left(1+\left(B_{0}+\varepsilon\right)(d(x))^{\mu}\right) \\ \times\left[g\left(\xi_{+} \psi(K(d(x)))\right)+\xi+\psi(K(d(x))) g^{\prime}\left(\Theta_{+}(d(x))\right)\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}\right] \\ \geq-\xi_{+}^{N}\left(\psi^{\prime}(K(d(x)))\right)^{N-1} \psi^{\prime \prime}(K(d(x))) k^{N+1}(d(x))(-\ln d(x))^{-\beta}\left\{(-\ln d(x))^{\beta}\right. \\ {l}\times\left[\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\left(\prod\limits_{i=1}^{N-1} \frac{\hat{m}_{+}}{\left(1-d(x) m_{+}\right)^{N-1}}-\hat{m}_{+}+\hat{m}_{+}\right)\right. \\ \left.-\frac{g(\xi+\psi(K(d(x))))}{\xi_{+}^{N} g(\psi(K(d(x))))}\right]+\left(C_{+}+\varepsilon\right)\left[N\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\right. \\ \times \prod\limits_{i=1}^{N-1} \frac{\kappa_{i}(\bar{x})}{1-d(x) \kappa_{i}(\bar{x})}-\frac{\xi_{+} \psi(K(d(x))) g^{\prime}\left(\xi_{+} \psi(K(d(x)))\right)}{g\left(\xi_{+} \psi(K(d(x)))\right)} \\ \left.\times \frac{g^{\prime}\left(\Theta_{+}(d(x))\right)}{g^{\prime}\left(\xi_{+} \psi(K(d(x)))\right)} \frac{g\left(\xi_{+} \psi(K(d(x)))\right)}{\xi_{+}^{N} g(\psi(K(d(x))))}\right]+\frac{\zeta_{+}(d(x))}{(-\ln d(x))^{-\beta}} \prod\limits_{i=1}^{N-1} \frac{\kappa_{i}(\bar{x})}{1-d(x) \kappa_{i}(\bar{x})} \\ -\left(B_{0}+\varepsilon\right) \frac{g\left(\xi_{+} \psi(K(d(x)))\right)}{\xi_{+}^{N} g(\psi(K(d(x))))}(d(x))^{\mu}(-\ln d(x))^{\beta}-\left(B_{0}+\varepsilon\right)\left(C_{+}+\varepsilon\right) \\ \left.\times \frac{\xi_{+} \psi(K(d(x))) g^{\prime}\left(\xi_{+} \psi(K(d(x)))\right)}{g(\xi+\psi(K(d(x))))} \frac{g^{\prime}\left(\Theta_{+}(d(x))\right)}{g^{\prime}\left(\xi_{+} \psi(K(d(x)))\right)} \frac{g\left(\xi_{+} \psi(K(d(x)))\right)}{\xi_{+}^{N} g(\psi(K(d(x))))}(d(x))^{\mu}\right\} \\ \geq-\xi_{+}^{N}\left(\psi^{\prime}(K(d(x)))\right)^{N-1} \psi^{\prime \prime}(K(d(x))) k^{N+1}(d(x))(-\ln d(x))^{-\beta} \sum_{i=1}^{3} I_{i+}(d(x))>0,\end{array} \end{equation}$$i.e., u_ε$\underline{u}_{\varepsilon}$is a subsolution of Eq. (1.1) in Ωδε$\Omega_{\delta_{\varepsilon}}$. Moreover, it follows by Lemma 4.3 that for i = 1, ..., NSiD2u_ε(x)=ξ+iψ′(K(d(x)))i1+C++ε(−lnd(x))−β+βC++εψ(K(d(x)))ψ′(K(d(x)))K(d(x))×K(d(x))k(d(x))d(x)(−lnd(x))−β−1iki(d(x))Siϵ1,⋯,ϵN−1−ξ+iψ′(K(d(x)))i−1ψ′′(K(d(x)))×ki+1(d(x))1+C++ε(−lnd(x))−β+βC++εψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)×(−lnd(x))−β−1i−1ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1×1+C++ε(−lnd(x))−β+2βC++εψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k(d(x))d(x)(−lnd(x))−β−1−βC++εψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))2(−lnd(x))−β−1×1−(β+1)(−lnd(x))−1Si−1ϵ1,⋯,ϵN−1$$\begin{equation}\begin{array}{l}&S_{i}\left(D^{2} \underline{u}_{\varepsilon}(x)\right)=\xi_{+}^{i}\left(\psi^{\prime}(K(d(x)))\right)^{i}\left(1+\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}+\beta\left(C_{+}+\varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))}\right. \\ & \left.\times \frac{K(d(x))}{k(d(x)) d(x)}(-\ln d(x))^{-\beta-1}\right)^{i} k^{i}(d(x)) S_{i}\left(\epsilon_{1}, \cdots, \epsilon_{N-1}\right)-\xi_{+}^{i}\left(\psi^{\prime}(K(d(x)))\right)^{i-1} \psi^{\prime \prime}(K(d(x))) \\ & \times k^{i+1}(d(x))\left(1+\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}+\beta\left(C_{+}+\varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{k(d(x)) d(x)}\right. \\ & \left.\times(-\ln d(x))^{-\beta-1}\right)^{i-1}\left[\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\right. \\ & \times\left(1+\left(C_{+}+\varepsilon\right)(-\ln d(x))^{-\beta}\right)+2 \beta\left(C_{+}+\varepsilon\right) \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{k(d(x)) d(x)}(-\ln d(x))^{-\beta-1} \\ & -\beta\left(C_{+}+\varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}(-\ln d(x))^{-\beta-1} \\ & \left.\times\left(1-(\beta+1)(-\ln d(x))^{-1}\right)\right] S_{i-1}\left(\epsilon_{1}, \cdots, \epsilon_{N-1}\right)\end{array} \end{equation}$$This implies that we can adjust the above positive constant δε$\delta_{\varepsilon}$such that for any x∈Ωδε$x \in \Omega_{\delta_{\varepsilon}}$S i ( D 2 u _ ε ( x ) ) > 0 for i = 1 , ⋅ ⋅ ⋅ , N . $$S_{i}(D^{2}\underline{u}_{\varepsilon}(x))>0 \,\mbox{for}\,i=1,\cdot\cdot\cdot,N.$$We obtain by Definition 4.4 that D2u_ε$D^{2} \underline{u}_{\varepsilon}$is positive definite in Ωδε.$\Omega_{\delta_{\varepsilon}}.$Letu‾ε(x)=−ξ−ψ(K(d(x)))(1+(C−−ε)(−lnd(x))−β),x∈Ωδε.$$\overline{u}_{\varepsilon}(x)=-\xi_{-}\psi(K(d(x)))\big(1+(C_{-}-\varepsilon)(-\lnd(x))^{-\beta}\big),\,x\in\Omega_{\delta_{\varepsilon}}.$$By the same calculation as (4.13), we obtaindetD2uˉε(x)−b(x)g−uˉε(x)≤−ξ−Nψ′(K(d(x)))N−1ψ′′(K(d(x)))kN+1(d(x))(−lnd(x))−β∑i=13Ii−(d(x))<0,$$\begin{equation}\begin{array}{l}\operatorname{det}\left(D^{2} \bar{u}_{\varepsilon}(x)\right)-b(x) g\left(-\bar{u}_{\varepsilon}(x)\right) \\ \leq-\xi_{-}^{N}\left(\psi^{\prime}(K(d(x)))\right)^{N-1} \psi^{\prime \prime}(K(d(x))) k^{N+1}(d(x))(-\ln d(x))^{-\beta} \sum_{i=1}^{3} I_{i-}(d(x))<0,\end{array} \end{equation}$$i.e. uˉε$\bar{u}_{\varepsilon}$is a supersolution of Eq. (1.1) inΩδε.$\Omega_{\delta_{\varepsilon}}.$Let u be the unique strictly convex solution to problem (1.1). Then, there exists a large constant M > 0 such thatu_ε(x)−Md(x)≤u(x)≤u‾ε(x)+Md(x),x∈{x∈Ω:d(x)=δε}.$$\underline{u}_{\varepsilon}(x)-Md(x)\leq u(x)\leq\overline{u}_{\varepsilon}(x)+Md(x),\,x\in\{x\in\Omega:d(x)=\delta_{\varepsilon}\}.$$We assert that(4.14)u(x)≥u_ε(x)−Md(x),x∈Ωδε$$\begin{equation}u(x)\geq\underline{u}_{\varepsilon}(x)-Md(x),\,x\in\Omega_{\delta_{\varepsilon}}\end{equation}$$and(4.15)u(x)≤u‾ε(x)+Md(x),x∈Ωδε.$$\begin{equation}u(x)\leq\overline{u}_{\varepsilon}(x)+Md(x),\,x\in\Omega_{\delta_{\varepsilon}}.\end{equation}$$Since D2u_ε$ D^{2}\underline{u}_{\varepsilon} $is positive definite in Ωδε$ \Omega_{\delta_{\varepsilon}} $and D2(−Md (x)) is positive semidefinite in Ωδε$ \Omega_{\delta_{\varepsilon}} $, we have by the Minkowski inequality that D2(u_ε(x)−Md(x))$ D^{2}(\underline{u}_{\varepsilon}(x)-Md(x)) $is positive definite in Ωδε$ \Omega_{\delta_{\varepsilon}} $anddet(D2(u_ε(x)−Md(x)))≥det(D2u_ε(x))≥b(x)g(−u_ε(x))≥b(x)g(−u_ε(x)+Md(x)),x∈Ωδε.$$\mbox{det}(D^{2}(\underline{u}_{\varepsilon}(x)-Md(x)))\geq\mbox{det}(D^{2}\underline{u}_{\varepsilon}(x))\geqb(x)g(-\underline{u}_{\varepsilon}(x))\geqb(x)g(-\underline{u}_{\varepsilon}(x)+Md(x)),\,x\in\Omega_{\delta_{\varepsilon}}.$$Similarly, we have D2(u(x)−Md (x)) is positive definite in Ωδε$ \Omega_{\delta_{\varepsilon}} $anddet(D2u(x)−Md(x))≥det(D2u(x))=b(x)g(−u(x))≥b(x)g(−u(x)+Md(x)).$$\mbox{det}(D^{2}u(x)-Md(x))\geq\mbox{det}(D^{2}u(x))=b(x)g(-u(x))\geq b(x)g(-u(x)+Md(x)).$$By Lemma 4.1, we see that (4.14)-(4.15) hold. Hence, for any x∈Ωδε$ x\in\Omega_{\delta_{\varepsilon}} $C++ε+Md(x)(−lnd(x))βξ+ψ(K(d(x)))≥−u(x)ξ+ψ(K(d(x)))−1(−lnd(x))β;C−−ε−Md(x)(−lnd(x))βξ−ψ(K(d(x)))≤−u(x)ξ−ψ(K(d(x)))−1(−lnd(x))β.$$\begin{equation}\begin{array}{l}C_{+}+\varepsilon+\frac{M d(x)(-\ln d(x))^{\beta}}{\xi+\psi(K(d(x)))} \geq\left(\frac{-u(x)}{\xi+\psi(K(d(x)))}-1\right)(-\ln d(x))^{\beta}; \\ C_{-}-\varepsilon-\frac{M d(x)(-\ln d(x))^{\beta}}{\xi_{-} \psi(K(d(x)))} \leq\left(\frac{-u(x)}{\xi-\psi(K(d(x)))}-1\right)(-\ln d(x))^{\beta}.\end{array} \end{equation}$$Since (γ+N)Dk−(N + 1)>0, we conclude from Lemma 3.1 (ii), Lemma 3.6 (iii), Proposition 2.7 and Proposition 2.5 (ii) thatC++ε≥lim supd(x)→0−u(x)ξ+ψ(K(d(x)))−1(−lnd(x))β;C−−ε≤lim infd(x)→0−u(x)ξ−ψ(K(d(x)))−1(−lnd(x))β.$$\begin{equation}\begin{array}{l}C_{+}+\varepsilon \geq \limsup _{d(x) \rightarrow 0}\left(\frac{-u(x)}{\xi+\psi(K(d(x)))}-1\right)(-\ln d(x))^{\beta}; \\ C_{-}-\varepsilon \leq \liminf _{d(x) \rightarrow 0}\left(\frac{-u(x)}{\xi_{-} \psi(K(d(x)))}-1\right)(-\ln d(x))^{\beta}.\end{array} \end{equation}$$Letting ε → 0, the proof is finished. □4.2Proof of Theorem 1.2Now, we prove Theorem 1.2. For fixed ε>0, we definew±(d(x))=ξ±ψ(K(d(x)))(1+(C˜±±ε)(d(x))μ),x∈Ωδ1,$$w_{\pm}(d(x))=\xi_{\pm}\psi(K(d(x)))\big(1+(\tilde{C}_{\pm}\pm\varepsilon)(d(x))^{\mu}\big),\,x\in\Omega_{\delta_{1}},$$where ξ ± and C˜±$ \tilde{C}_{\pm} $are given in Theorem 1.2. By the Lagrange’s mean value theorem, we see that there exist λ ± ∈ (0, 1) and(4.16)Θ±(d(x))=ξ±ψ(K(d(x)))(1+λ±(C˜±±ε)(d(x))μ)$$\begin{equation}\Theta_{\pm}(d(x))=\xi_{\pm}\psi(K(d(x)))\big(1+\lambda_{\pm}(\tilde{C}_{\pm}\pm\varepsilon)(d(x))^{\mu}\big)\end{equation}$$such that for any x∈Ωδ1$ x\in\Omega_{\delta_{1}} $g(w±(d(x)))=g(ξ±ψ(K(d(x))))+ξ±ψ(K(d(x)))g′(Θ±(d(x)))(C±±ε)(d(x))μ.$$\begin{split}g(w_{\pm}(d(x)))&=g(\xi_{\pm}\psi(K(d(x))))+\xi_{\pm}\psi(K(d(x)))g'(\Theta_{\pm}(d(x)))(C_{\pm}\pm\varepsilon)(d(x))^{\mu}.\end{split}$$By Proposition 2.2, we see that (4.2) still holds. Moreover, we can adjust δ1 > 0 such that (4.3) holds here.Proof. Similar to the proof of Theorem 1.1, the proof is divided into the following two steps.Step 1. We first define functions I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as follows.For fixed ε > 0 and ∀x∈Ωδ1$ \forall\,x\in\Omega_{\delta_{1}} $, we defineI 1 ± ( d ( x ) ) = 1 ( d ( x ) ) μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 m ^ ± − g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ; I 2 ± ( d ( x ) ) = C ~ ± ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × N + μ ( N − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + 2 μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + μ × ( μ − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 B ± ∏ i = 1 N − 1 1 − d ( x ) κ i ( x ¯ ) − 1 − ξ ± ψ ( K ( d ( x ) ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) g ′ Θ ± ( d ( x ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ± ε ψ ′ ( x ) ( x ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × N + μ ( N − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + 2 μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) + μ × ( μ − 1 ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 × m ^ + ∏ i = 1 N − 1 1 − d ( x ) κ i ( x ¯ ) − 1 − ξ ± ψ ( K ( d ( x ) ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) g ′ Θ ± ( d ( x ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) × g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) − B 0 ± ε g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ; I 3 ± ( d ( x ) ) = ( d ( x ) ) − μ ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) + 1 × m ^ ± ∑ i = 1 N − 1 C N − 1 i ( − 1 ) i + 1 m ± d ( x ) i 1 − d ( x ) m ± N − 1 − B 0 ± ε C ~ ± ± ε ξ ± ψ ( K ( d ( x ) ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) × g ′ Θ ± ( d ( x ) ) g ′ ξ ± ψ ( K ( d ( x ) ) ) g ξ ± ψ ( K ( d ( x ) ) ) ξ ± N g ( ψ ( K ( d ( x ) ) ) ) ( d ( x ) ) μ + ζ ~ ± ( d ( x ) ) ( d ( x ) ) μ ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) k ( x ¯ ) , $$\begin{equation}\begin{array}{l}I_{1 \pm}(d(x))=\frac{1}{(d(x))^{\mu}}\left[\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right) \hat{m}_{\pm}-\frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))}\right]; \\ I_{2 \pm}(d(x))=\tilde{C}_{\pm}\left\{\left[\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\right.\right. \\ \times\left(N+\mu(N-1) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}\right)+\frac{2 \mu \psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}+\mu \\ \left.\times(\mu-1) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}\right] \mathscr{B}_{\pm} \prod\limits_{i=1}^{N-1}\left(1-d(x) \kappa_{i}(\bar{x})\right)^{-1} \\ \left.-\frac{\xi_{\pm} \psi(K(d(x))) g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)}{g\left(\xi_{\pm} \psi(K(d(x)))\right)} \frac{g^{\prime}\left(\Theta_{\pm}(d(x))\right)}{g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)} \frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))}\right\}\\ \pm \varepsilon\left\{\left[\left(\frac{\left.\left.\psi^{\prime}(x)(x)\right)\right)}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right)\right.\right. \\ \times\left(N+\mu(N-1) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}\right)+\frac{2 \mu \psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}+\mu \\ \left.\times(\mu-1) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}\right] \\ \times \hat{m}_{+} \prod\limits_{i=1}^{N-1}\left(1-d(x) \kappa_{i}(\bar{x})\right)^{-1}-\frac{\xi_{\pm} \psi(K(d(x))) g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)}{g\left(\xi_{\pm} \psi(K(d(x)))\right)} \frac{g^{\prime}\left(\Theta_{\pm}(d(x))\right)}{g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)} \\ \left.\times \frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))}\right\}-\left(B_{0} \pm \varepsilon\right) \frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))};\\ I_{3 \pm}(d(x))=(d(x))^{-\mu}\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}+1\right) \\ \times \frac{\hat{m}_{\pm} \sum_{i=1}^{N-1} C_{N-1}^{i}(-1)^{i+1}\left(m_{\pm} d(x)\right)^{i}}{\left(1-d(x) m_{\pm}\right)^{N-1}}-\left(B_{0} \pm \varepsilon\right)\left(\tilde{C}_{\pm} \pm \varepsilon\right) \frac{\xi_{\pm} \psi(K(d(x))) g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)}{g\left(\xi_{\pm} \psi(K(d(x)))\right)} \\ \times \frac{g^{\prime}\left(\Theta_{\pm}(d(x))\right)}{g^{\prime}\left(\xi_{\pm} \psi(K(d(x)))\right)} \frac{g\left(\xi_{\pm} \psi(K(d(x)))\right)}{\xi_{\pm}^{N} g(\psi(K(d(x))))}(d(x))^{\mu}+\frac{\tilde{\zeta}_{\pm}(d(x))}{(d(x))^{\mu}} \prod\limits_{i=1}^{N-1} \frac{\kappa_{i}(\bar{x})}{1-d(x)_{k}(\bar{x})},\end{array} \end{equation}$$whereB±=mˆ±,if B0≥0,B±=mˆ∓, if B0<0,$$\begin{equation}\left\{\begin{array}{ll}\mathscr{B}_{\pm}=\hat{m}_{\pm}, & \text {if } B_{0} \geq 0, \\ \mathscr{B}_{\pm}=\hat{m}_{\mp}, & \text { if } B_{0}<0,\end{array}\right. \end{equation}$$m ± are defined by (4.6) andζ ~ ± ( d ( x ) ) = C ~ ± ± ε 2 ( N − 1 ) 1 + ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) × 1 + μ ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) ( d ( x ) ) 2 μ + 2 μ C ~ ± ± ε 2 ( N − 1 ) 1 + μ ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) × K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) ( d ( x ) ) 2 μ + μ ( μ − 1 ) C ~ ± ± ε 2 ( N − 1 ) 1 + μ ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) × K ( d ( x ) ) d ( x ) k ( d ( x ) ) ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 ( d ( x ) ) 2 μ + v ~ ± ( d ( x ) ) × 1 + ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) k ′ ( d ( x ) ) k 2 ( d ( x ) ) 1 + C ~ ± ± ε ( d ( x ) ) μ + 2 μ C ~ ± ± ε ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) ( d ( x ) ) μ + μ ( μ − 1 ) C ~ ± ± ε ψ ( K ( d ( x ) ) ) ψ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) ψ ′ ( K ( d ( x ) ) ) ψ ′ ′ ( K ( d ( x ) ) ) K ( d ( x ) ) K ( d ( x ) ) d ( x ) k ( d ( x ) ) 2 ( d ( x ) ) μ $$ \begin{equation}\begin{array}{l}\tilde{\zeta}_{\pm}(d(x))=\left(\tilde{C}_{\pm} \pm \varepsilon\right)^{2}(N-1)\left(1+\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}\right) \\ \times\left(1+\frac{\mu \psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}\right)(d(x))^{2 \mu} \\ +2 \mu\left(\tilde{C}_{\pm} \pm \varepsilon\right)^{2}(N-1)\left(1+\frac{\mu \psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}\right) \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \\ \times \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}(d(x))^{2 \mu}+\mu(\mu-1)\left(\tilde{C}_{\pm} \pm \varepsilon\right)^{2}(N-1)\left(1+\frac{\mu \psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))}\right.\\ \left.\times \frac{K(d(x))}{d(x) k(d(x))}\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}(d(x))^{2 \mu}+\tilde{v}_{\pm}(d(x)) \\ \times\left[\left(1+\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))}\right)\left(1+\left(\tilde{C}_{\pm} \pm \varepsilon\right)(d(x))^{\mu}\right)\right. \\ +2 \mu\left(\tilde{C}_{\pm} \pm \varepsilon\right) \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))} \frac{K(d(x))}{d(x) k(d(x))}(d(x))^{\mu} \\ \left.+\mu(\mu-1)\left(\tilde{C}_{\pm} \pm \varepsilon\right) \frac{\psi(K(d(x)))}{\psi^{\prime}(K(d(x))) K(d(x))} \frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}\left(\frac{K(d(x))}{d(x) k(d(x))}\right)^{2}(d(x))^{\mu}\right]\end{array} \end{equation}$$withν˜±(d(x))=∑i=2N−1CN−1i(R˜±(d(x)))i,$$\tilde{\nu}_{\pm}(d(x))=\sum_{i=2}^{N-1}C_{N-1}^{i}(\tilde{\mathfrak{R}}_{\pm}(d(x)))^{i},$$where CN−1i$ C_{N-1}^{i} $is given in (4.7) andR˜±(d(x))=(C˜±±ε)(1+μψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x)))(d(x))μ.$$\tilde{\mathfrak{R}}_{\pm}(d(x))=(\tilde{C}_{\pm}\pm\varepsilon)\bigg(1+\frac{\mu\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}\bigg)(d(x))^{\mu}.$$Next, we prove(4.17)limd(x)→0(I1±(d(x))+I2±(d(x))+I3±(d(x)))=±εϱ±mˆ±,$$\begin{equation}\lim_{d(x)\rightarrow0}(I_{1\pm}(d(x))+I_{2\pm}(d(x))+I_{3\pm}(d(x)))=\pm\varepsilon\varrho_{\pm}\hat{m}_{\pm},\end{equation}$$where(4.18)ϱ±=(1−μ)(γ+N)Dk−(N+1)γ−1N+μ(N−1)γ+NN+1Dk+μγ+Nγ−1DkN+μ(N−1)γ+NN+1Dk−2+μ(1−μ)(γ+N)2(γ−1)(N+1)Dk2mˆ+mˆ±+(γ+N)Dk−(N+1).$$\begin{equation}\begin{array}{l}\varrho_{\pm}=\left[\frac{(1-\mu)(\gamma+N) D_{k}-(N+1)}{\gamma-1}\left(N+\mu(N-1) \frac{\gamma+N}{N+1} D_{k}\right)\right. \\ \left.+\mu \frac{\gamma+N}{\gamma-1} D_{k}\left(N+\mu(N-1) \frac{\gamma+N}{N+1} D_{k}-2\right)+\mu(1-\mu) \frac{(\gamma+N)^{2}}{(\gamma-1)(N+1)} D_{k}^{2}\right] \frac{\hat{m}_{+}}{\hat{m}_{\pm}} \\ +(\gamma+N) D_{k}-(N+1).\end{array} \end{equation}$$To prove (4.17), we calculate the limits of I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as d(x)→0.$d(x) \rightarrow 0.$• First, we investigate the limits of I1±(d(x)) as d(x)→0.$I_{1 \pm}(d(x)) \text { as } d(x) \rightarrow 0.$As the preliminaries, we prove some limits as below.Case 1. When Dk ∈ (0, 1), by a direct calculation, we haveN+1(1−μ)Dk−N>N+1(1−μ)−N>(1+μ)N+1N+1−μ.$$\frac{N+1}{(1-\mu)D_{k}}-N>\frac{N+1}{(1-\mu)}-N>\frac{(1+\mu)N+1}{N+1-\mu}.$$So, by (1.15), Lemma 3.1 (ii) and Lemma 3.6 (viii) and (x) with ρ=μ ∈ (0, 1), we obtain(4.19)limd(x)→0+(d(x))−μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1K(d(x))k′(d(x))k2(d(x))mˆ±=limd(x)→0+K(d(x))d(x)μ(K(d(x)))−μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1K(d(x))k′(d(x))k2(d(x))mˆ±=0, if S2 and g3 hold with θ(N+1)μ−N>γ>N+1(1−μ)Dk−N,0, if s1 holds with γ>N+1(1−μ)Dk−N in g2$$\begin{equation}\begin{aligned} & \lim _{d(x) \rightarrow 0^{+}}(d(x))^{-\mu}\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}+\frac{\gamma+N}{\gamma-1}\right) \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))} \hat{m}_{\pm} \\=& \lim _{d(x) \rightarrow 0^{+}}\left(\frac{K(d(x))}{d(x)}\right)^{\mu}(K(d(x)))^{-\mu}\left(\frac{\psi^{\prime}(K(d(x)))}{\psi^{\prime \prime}(K(d(x))) K(d(x))}+\frac{\gamma+N}{\gamma-1}\right) \frac{K(d(x)) k^{\prime}(d(x))}{k^{2}(d(x))} \hat{m}_{\pm} \\=&\left\{\begin{array}{ll}0, & \text { if }\left(\mathbf{S}_{2}\right) \text { and }\left(\mathbf{g}_{3}\right) \text { hold with } \frac{\theta(N+1)}{\mu}-N>\gamma>\frac{N+1}{(1-\mu) D_{k}}-N, \\ 0, & \text { if }\left(\mathbf{s}_{1}\right) \text { holds with } \gamma>\frac{N+1}{(1-\mu) D_{k}}-N \text { in }\left(\mathbf{g}_{2}\right)\end{array}\right.\end{aligned} \end{equation}$$and(4.20)limd(x)→0(d(x))−μ(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=limd(x)→0(K(d(x))d(x))μ(K(d(x)))−μ(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=0, if (S2) and (g3) hold with θ(N+1)μ−N>γ,0, if (S1) holds in (g2).$$\begin{equation}\begin{split}&\lim\limits_{d(x)\rightarrow0}(d(x))^{-\mu}\bigg(\xi_{\pm}^{-(\gamma+N)}-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg)\\=&\lim\limits_{d(x)\rightarrow0}\bigg(\frac{K(d(x))}{d(x)}\bigg)^{\mu}(K(d(x)))^{-\mu}\bigg(\xi_{\pm}^{-(\gamma+N)}-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg)\\=&\begin{cases} 0, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and}\mathbf{(g_{3})} \mbox{ hold with }\frac{\theta(N+1)}{\mu}-N>\gamma,\\0, &\mbox{ if }\mathbf{(S_{1})}\mbox{ holds} \mbox{ in}\mathbf{(g_{2})}.\end{cases}\end{split}\end{equation}$$Case 2. When Dk ∈ [1, ∞), by (1.15), Lemma 3.1 (ii) and Lemma 3.6 (viii) and (x) with ρ=1, we obtain(4.21)limd(x)→0(d(x))−μ(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1)K(d(x))k′(d(x))k2(d(x))mˆ±=limd(x)→0K(d(x))(d(x))μ(K(d(x)))−1(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))+γ+Nγ−1)K(d(x))k′(d(x))k2(d(x))mˆ±=0, if (S2) and (g3) hold with θ(N+1)−N>γ>2N+1N,0, if (S1) holds with γ>2N+1N in (g2)$$\begin{equation}\begin{split}&\lim\limits_{d(x)\rightarrow0}(d(x))^{-\mu}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}+\frac{\gamma+N}{\gamma-1}\bigg)\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}\hat{m}_{\pm}\\=&\lim\limits_{d(x)\rightarrow0}\frac{K(d(x))}{(d(x))^{\mu}}(K(d(x)))^{-1}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}+\frac{\gamma+N}{\gamma-1}\bigg)\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}\hat{m}_{\pm}\\=&\begin{cases} 0, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and}\mathbf{(g_{3})} \mbox{ hold with }\theta(N+1)-N>\gamma>\frac{2N+1}{N},\\0, &\mbox{ if }\mathbf{(S_{1})}\mbox{ holds with }\gamma>\frac{2N+1}{N} \mbox{ in }\mathbf{(g_{2})}\end{cases}\end{split}\end{equation}$$and(4.22)limd(x)→0(d(x))−μ(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=limd(x)→0K(d(x))(d(x))μ(K(d(x)))−1(ξ±−(γ+N)−g(ξ±ψ(K(d(x))))ξ±Ng(ψ(K(d(x)))))=0, if (S2) and (g3) hold with θ(N+1)−N>γ,0, if (S1) holds in (g2).$$\begin{equation}\begin{split}&\lim\limits_{d(x)\rightarrow0}(d(x))^{-\mu}\bigg(\xi_{\pm}^{-(\gamma+N)}-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg)\\=&\lim\limits_{d(x)\rightarrow0}\frac{K(d(x))}{(d(x))^{\mu}}(K(d(x)))^{-1}\bigg(\xi_{\pm}^{-(\gamma+N)}-\frac{g(\xi_{\pm}\psi(K(d(x))))}{\xi_{\pm}^{N}g(\psi(K(d(x))))}\bigg)\\=&\begin{cases} 0, &\mbox{ if }\mathbf{(S_{2})} \mbox{ and}\mathbf{(g_{3})} \mbox{ hold with }\theta(N+1)-N>\gamma,\\0, &\mbox{ if }\mathbf{(S_{1})}\mbox{ holds} \mbox{ in}\mathbf{(g_{2})}.\end{cases}\end{split}\end{equation}$$On the other hand, a simple calculation shows that(4.23)−limd(x)→0γ+Nγ−1(d(x))−μ(K(d(x))k′(d(x))k2(d(x))−(1−Dk))mˆ±=−limd(x)→0γ+Nγ−1(d(x))1−μ(d(x))−1(K(d(x))k′(d(x))k2(d(x))−(1−Dk))mˆ±=0$$\begin{equation}\begin{split}-&\lim\limits_{d(x)\rightarrow0}\frac{\gamma+N}{\gamma-1}(d(x))^{-\mu}\bigg(\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}-(1-D_{k})\bigg)\hat{m}_{\pm}\\=-&\lim\limits_{d(x)\rightarrow0}\frac{\gamma+N}{\gamma-1}(d(x))^{1-\mu}(d(x))^{-1}\bigg(\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}-(1-D_{k})\bigg)\hat{m}_{\pm}=0\end{split}\end{equation}$$We obtain by combining (4.19)-(4.20) (or (4.21)-(4.22)) and (4.23) that(4.24)limd(x)→0I1±(d(x))=0.$$\begin{equation}\lim\limits_{d(x)\rightarrow0}I_{1\pm}(d(x))=0.\end{equation}$$• Second, by (4.2), Lemma 3.1 (ii), Lemma 3.6 (iii)-(iv) and the choices of ξ ± , C˜±$ \tilde{C}_{\pm} $in Theorem 1.2, we obtain(4.25)limd(x)→0I2±(d(x))=C˜±[((1−μ)(γ+N)Dk−(N+1)γ−1(N+μ(N−1)γ+NN+1Dk)+μγ+Nγ−1Dk(N+μ(N−1)γ+NN+1Dk−2)+μ(1−μ)(γ+N)2(γ−1)(N+1)Dk2)B±+γ((γ+N)Dk−(N+1))mˆ±γ−1]−B0((γ+N)Dk−(N+1))mˆ±γ−1±εϱ±mˆ±=±εϱ±mˆ±, where ϱ± are given by (4.18).$$\begin{equation}\begin{split}&\lim\limits_{d(x)\rightarrow0}I_{2\pm}(d(x))=\tilde{C}_{\pm}\bigg[\bigg(\frac{(1-\mu)(\gamma+N)D_{k}-(N+1)}{\gamma-1}\bigg(N+\mu(N-1)\frac{\gamma+N}{N+1}D_{k}\bigg)\\&+\mu\frac{\gamma+N}{\gamma-1}D_{k}\bigg(N+\mu(N-1)\frac{\gamma+N}{N+1}D_{k}-2\bigg)+\mu(1-\mu)\frac{(\gamma+N)^{2}}{(\gamma-1)(N+1)}D_{k}^{2}\bigg)\mathscr{B}_{\pm}\\&+\frac{\gamma\big((\gamma+N)D_{k}-(N+1)\big)\hat{m}_{\pm}}{\gamma-1}\bigg]-B_{0}\frac{\big((\gamma+N)D_{k}-(N+1)\big)\hat{m}_{\pm}}{\gamma-1}\\&\pm\varepsilon\varrho_{\pm}\hat{m}_{\pm}=\pm\varepsilon\varrho_{\pm} \hat{m}_{\pm},\mbox{ where} \varrho_{\pm} \mbox{ are given by } (4.18).\end{split}\end{equation}$$• Third, we conclude by Lemma 3.1 (ii) Lemma 3.6 (iii)-(iv) and a straightforward calculation that(4.26)limd(x)→0I3±(d(x))=0.$$\begin{equation}\lim\limits_{d(x)\rightarrow0}I_{3\pm}(d(x))=0.\end{equation}$$Combining (4.24) and (4.25)-(4.26), we obtain (4.17) holds. Moreover, it follows by (1.15) that thatϱ ± >(γ+N)Dk−(N + 1)>0. By (b1)-(b2) and (4.17), we see that there exists a sufficiently small constant δε > 0 such that (4.12) and (4.4)-(4.5) hold here.Step 2. Letu_ε(x)=−ξ+ψ(K(d(x)))(1+(C˜++ε)(d(x))μ),x∈Ωδε.$$\underline{u}_{\varepsilon}(x)=-\xi_{+}\psi(K(d(x)))\big(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}\big),\,x\in\Omega_{\delta_{\varepsilon}}.$$Theng(−u_ε(x))=g(ξ+ψ(K(d(x))))+ξ+ψ(K(d(x)))g′(Θ+(d(x)))(C˜++ε)(d(x))μ,x∈Ωδε,$$g(-\underline{u}_{\varepsilon}(x))=g(\xi_{+}\psi(K(d(x))))+\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu},\,x\in\Omega_{\delta_{\varepsilon}},$$where Θ+(d(x)) is given by (4.16). By Lemma 4.2, we have for any x∈Ωδε(4.27)det(D2u_ε(x))−b(x)g(−u_ε(x))≥−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))×(1+(C˜++ε)(N−1)(d(x))μ+μ(C˜++ε)(N−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ+ν˜+(d(x)))[(1+ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x)))(1+(C˜++ε)(d(x))μ)+2μ(C++ε)ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ+μ(μ−1)(C˜++ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2(d(x))μ]×∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)−kN+1(d(x))(1+(B0+ε)(d(x))μ)[g(ξ+ψ(K(d(x))))+ξ+ψ(K(d(x)))g′(Θ+(d(x)))(C˜++ε)(d(x))μ]=−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ×{(d(x))−μ(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)+(C˜++ε)[(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(N+μ(N−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))×K(d(x))d(x)k(d(x)))+2μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))+μ(μ−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))×ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2]∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)+ζ˜+(d(x))(d(x))μ∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)}−kN+1(d(x))g(ξ+ψ(K(d(x))))−(B0+ε)kN+1(d(x))g(ξ+ψ(K(d(x))))(d(x))μ−(C˜++ε)kN+1(d(x))ξ+ψ(K(d(x)))g′(Θ+(d(x)))(d(x))μ−(B0+ε)(C˜++ε)kN+1(d(x))×ξ+ψ(K(d(x)))g′(Θ+(d(x)))(d(x))2μ=−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ×{(d(x))−μ[(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ+−g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))]+(C˜++ε)[((ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)(N+μ(N−1)×ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x)))+2μψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))+μ(μ−1)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2)∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)−ξ+ψ(K(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))g′(Θ+(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))]−(B0+ε)g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))−(B0+ε)(C˜++ε)ξ+ψ(K(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))×g′(Θ+(d(x)))g′(ξ+ψ(K(d(x))))g(ξ+ψ(K(d(x))))ξ+Ng(ψ(K(d(x))))(d(x))μ+(d(x))−μ(ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x))+1)mˆ+∑i=1N−1CN−1i(−1)i+1(m+d(x))i(1−d(x)m+)N−1+ζ˜+(d(x))(d(x))μ∏i=1N−1κi(xˉ)1−d(x)κi(xˉ)}≥−ξ+N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ∑i=13Ii+(d(x))>0,$$\begin{align}\nonumber&\mbox{det}(D^{2}\underline{u}_{\varepsilon}(x))-b(x)g(-\underline{u}_{\varepsilon}(x))\geq-\xi_{+}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))\\\nonumber&\times\bigg(1+(\tilde{C}_{+}+\varepsilon)(N-1)(d(x))^{\mu}+\mu(\tilde{C}_{+}+\varepsilon)(N-1)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}(d(x))^{\mu}\\\nonumber&+\tilde{\nu}_{+}(d(x))\bigg)\bigg[\bigg(1+\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}\bigg)\big(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}\big)\\\nonumber&+2\mu(C_{+}+\varepsilon)\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}(d(x))^{\mu}\\\nonumber&+\mu(\mu-1)(\tilde{C}_{+}+\varepsilon)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}(d(x))^{\mu}\bigg]\\\nonumber&\times\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}-k^{N+1}(d(x))\big(1+(B_{0}+\varepsilon)(d(x))^{\mu}\big)\big[g(\xi_{+}\psi(K(d(x))))\\\nonumber&+\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}\big]\\\nonumber&=-\xi_{+}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))(d(x))^{\mu}\\\nonumber&\times\bigg\{(d(x))^{-\mu}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\\\nonumber&+(\tilde{C}_{+}+\varepsilon)\bigg[\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\bigg(N+\mu(N-1)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\\\nonumber&\times\frac{K(d(x))}{d(x)k(d(x))}\bigg)+\frac{2\mu\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}+\frac{\mu(\mu-1)\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\\\nonumber&\times\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}\bigg]\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}+\frac{\tilde{\zeta}_{+}(d(x))}{(d(x))^{\mu}}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\bigg\}\\\nonumber&-k^{N+1}(d(x))g(\xi_{+}\psi(K(d(x))))-(B_{0}+\varepsilon)k^{N+1}(d(x))g(\xi_{+}\psi(K(d(x))))(d(x))^{\mu}\\\nonumber&-(\tilde{C}_{+}+\varepsilon)k^{N+1}(d(x))\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(d(x))^{\mu}-(B_{0}+\varepsilon)(\tilde{C}_{+}+\varepsilon)k^{N+1}(d(x))\\\nonumber&\times\xi_{+}\psi(K(d(x)))g'(\Theta_{+}(d(x)))(d(x))^{2\mu}\\\nonumber&=-\xi_{+}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))(d(x))^{\mu}\\\nonumber&\times\bigg\{(d(x))^{-\mu}\bigg[\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\hat{m}_{+}-\frac{g(\xi_{+}\psi(K(d(x))))}{\xi_{+}^{N}g(\psi(K(d(x))))}\bigg]\\\nonumber&+(\tilde{C}_{+}+\varepsilon)\bigg[\bigg(\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\bigg(N+\mu(N-1)\\\label{4.27l}&\times\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}\bigg)+\frac{2\mu\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}\\\nonumber&+\frac{\mu(\mu-1)\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}\bigg)\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\\\nonumber&-\frac{\xi_{+}\psi(K(d(x)))g'(\xi_{+}\psi(K(d(x))))}{g(\xi_{+}\psi(K(d(x))))}\frac{g'(\Theta_{+}(d(x)))}{g'(\xi_{+}\psi(K(d(x))))}\frac{g(\xi_{+}\psi(K(d(x))))}{\xi_{+}^{N}g(\psi(K(d(x))))}\bigg]\\\nonumber&-(B_{0}+\varepsilon)\frac{g(\xi_{+}\psi(K(d(x))))}{\xi_{+}^{N}g(\psi(K(d(x))))}-(B_{0}+\varepsilon)(\tilde{C}_{+}+\varepsilon)\frac{\xi_{+}\psi(K(d(x)))g'(\xi_{+}\psi(K(d(x))))}{g(\xi_{+}\psi(K(d(x))))}\\\nonumber&\times\frac{g'(\Theta_{+}(d(x)))}{g'(\xi_{+}\psi(K(d(x))))}\frac{g(\xi_{+}\psi(K(d(x))))}{\xi_{+}^{N}g(\psi(K(d(x))))}(d(x))^{\mu}\\\nonumber&+(d(x))^{-\mu}\bigg(\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}+1\bigg)\frac{\hat{m}_{+}\sum\limits_{i=1}^{N-1}C_{N-1}^{i}(-1)^{i+1}(m_{+}d(x))^{i}}{(1-d(x)m_{+})^{N-1}}\\\nonumber&+\frac{\tilde{\zeta}_{+}(d(x))}{(d(x))^{\mu}}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\bigg\}\\&\geq-\xi_{+}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))(d(x))^{\mu}\sum\limits_{i=1}^{3}I_{i+}(d(x))>0, \nonumber\end{align}$$i.e., uε is a subsolution of Eq. (1.1) in Ωδε. Moreover, it follows by Lemma 4.3 that for i=1, ... ,NSi(D2u_ε(x))=ξ+i(ψ′(K(d(x))))iki(d(x))(1+(C˜++ε)(d(x))μ+μ(C˜++ε)K(d(x))d(x)k(d(x))×ψ(K(d(x)))ψ′(K(d(x)))K(d(x))(d(x))μ)iSi(ϵ1,⋅⋅⋅,ϵN−1)−ξ+i(ψ′(K(d(x))))i−1ψ′′(K(d(x)))ki+1(d(x))(1+(C˜++ε)(d(x))μ+μ(C˜++ε)×ψ(K(d(x)))ψ′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ)i−1[(1+ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))k′(d(x))k2(d(x)))×(1+(C˜++ε)(d(x))μ)+2μ(C++ε)ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))K(d(x))d(x)k(d(x))(d(x))μ+μ(μ−1)(C˜++ε)ψ(K(d(x)))ψ′(K(d(x)))K(d(x))ψ′(K(d(x)))ψ′′(K(d(x)))K(d(x))(K(d(x))d(x)k(d(x)))2(d(x))μ]×Si−1(ϵ1,⋅⋅⋅,ϵN−1).$$\begin{split}&S_{i}(D^{2}\underline{u}_{\varepsilon}(x))=\xi_{+}^{i}(\psi'(K(d(x))))^{i}k^{i}(d(x))\bigg(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}+\mu(\tilde{C}_{+}+\varepsilon)\frac{K(d(x))}{d(x)k(d(x))}\\&\times\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}(d(x))^{\mu}\bigg)^{i}S_{i}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1})\\&-\xi_{+}^{i}(\psi'(K(d(x))))^{i-1}\psi''(K(d(x)))k^{i+1}(d(x))\bigg(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}+\mu(\tilde{C}_{+}+\varepsilon)\\&\times\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}(d(x))^{\mu}\bigg)^{i-1}\bigg[\bigg(1+\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))k'(d(x))}{k^{2}(d(x))}\bigg)\\&\times\big(1+(\tilde{C}_{+}+\varepsilon)(d(x))^{\mu}\big)+2\mu(C_{+}+\varepsilon)\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\frac{K(d(x))}{d(x)k(d(x))}(d(x))^{\mu}\\&+\mu(\mu-1)(\tilde{C}_{+}+\varepsilon)\frac{\psi(K(d(x)))}{\psi'(K(d(x)))K(d(x))}\frac{\psi'(K(d(x)))}{\psi''(K(d(x)))K(d(x))}\bigg(\frac{K(d(x))}{d(x)k(d(x))}\bigg)^{2}(d(x))^{\mu}\bigg]\\&\timesS_{i-1}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1}).\end{split}\quad\quad\quad\quad\quad\,$$This implies that we can adjust the above positive constant δε such that for any x∈ΩδεS i ( D 2 u _ ε ( x ) ) > 0 for i = 1 , ⋅ ⋅ ⋅ , N . $$S_{i}(D^{2}\underline{u}_{\varepsilon}(x))>0 \,\mbox{for}\,i=1,\cdot\cdot\cdot,N.$$We obtain by Lemma 4.4 that D2uε is positive definite in Ωδε.Letu‾ε(x)=−ξ−ψ(K(d(x)))(1+(C˜−−ε)(d(x))μ),x∈Ωδε$$\begin{split}&\overline{u}_{\varepsilon}(x)=-\xi_{-}\psi(K(d(x)))\big(1+(\tilde{C}_{-}-\varepsilon)(d(x))^{\mu}\big),\,x\in\Omega_{\delta_{\varepsilon}}\end{split}$$By the same calculation as (4.27), we obtaindet(D2u‾ε(x))−b(x)g(−u‾ε(x))≤−ξ−N(ψ′(K(d(x))))N−1ψ′′(K(d(x)))kN+1(d(x))(d(x))μ∑i=13Ii−(d(x))<0,$$\begin{split}&\mbox{det}(D^{2}\overline{u}_{\varepsilon}(x))-b(x)g(-\overline{u}_{\varepsilon}(x))\\&\leq-\xi_{-}^{N}(\psi'(K(d(x))))^{N-1}\psi''(K(d(x)))k^{N+1}(d(x))(d(x))^{\mu}\sum_{i=1}^{3}I_{i-}(d(x))<0,\end{split}$$i.e. uε is a supersolution of Eq. (1.1) in Ωδε.Let u be the unique strictly convex solution to problem (1.1). Through the same argument as Theorem 1.1, we see that there exists a large constant M > 0 such thatu_ε(x)−Md(x)≤u(x)≤u‾ε(x)+Md(x),x∈Ωδε.$$\underline{u}_{\varepsilon}(x)-Md(x)\leq u(x)\leq\overline{u}_{\varepsilon}(x)+Md(x),\,x\in\Omega_{\delta_{\varepsilon}}.$$Hence, for any x∈ ΩδεC˜++ε+M(d(x))1−μξ+ψ(K(d(x)))≥(−u(x)ξ+ψ(K(d(x)))−1)(d(x))−μ;C˜−−ε−M(d(x))1−μξ−ψ(K(d(x)))≤(−u(x)ξ−ψ(K(d(x)))−1)(d(x))−μ.$$\begin{split}\tilde{C}_{+}+\varepsilon+\frac{M(d(x))^{1-\mu}}{\xi_{+}\psi(K(d(x)))}&\geq\bigg(\frac{-u(x)}{\xi_{+}\psi(K(d(x)))}-1\bigg)(d(x))^{-\mu};\\\tilde{C}_{-}-\varepsilon-\frac{M(d(x))^{1-\mu}}{\xi_{-}\psi(K(d(x)))}&\leq\bigg(\frac{-u(x)}{\xi_{-}\psi(K(d(x)))}-1\bigg)(d(x))^{-\mu}.\end{split}$$Since (1-μ)(γ+N)Dk-(N+1)>0, we conclude from Lemma 3.1 (ii) , Lemma 3.6 (iii), Proposition 2.7 and Proposition 2.5 (ii) thatC˜++ε≥lim supd(x)→0(−u(x)ξ+ψ(K(d(x)))−1)(d(x))−μ;C˜−−ε≤lim infd(x)→0(−u(x)ξ−ψ(K(d(x)))−1)(d(x))−μ.$$\begin{split}\tilde{C}_{+}+\varepsilon&\geq\limsup_{d(x)\rightarrow0}\bigg(\frac{-u(x)}{\xi_{+}\psi(K(d(x)))}-1\bigg)(d(x))^{-\mu};\\\tilde{C}_{-}-\varepsilon&\leq\liminf_{d(x)\rightarrow0}\bigg(\frac{-u(x)}{\xi_{-}\psi(K(d(x)))}-1\bigg)(d(x))^{-\mu}.\end{split}$$Letting ε → 0, the proof is finished. □4.3Proof of Theorem 1.3Now, we prove Theorem 1.3. As before, for fixed ε > 0, we definew ± ( d ( x ) ) = η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + ( C ± ∗ ± ε ) M ( d ( x ) ) ) , x ∈ Ω δ 1 , $$w_{\pm}(d(x))=\eta_{\pm}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x))\big),\,x\in\Omega_{\delta_{1}},$$where η ± , C ± * and M$ \mathfrak{M} $are given in Theorem 1.3. By the Lagrange’s mean value theorem, it is clear that there exist λ ± ∈(0, 1) and(4.28)Θ ± ( d ( x ) ) = η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ( 1 + λ ± ( C ± ∗ ± ε ) M ( d ( x ) ) ) $$\begin{equation}\Theta_{\pm}(d(x))=\eta_{\pm}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\big(1+\lambda_{\pm}(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x))\big)\end{equation}$$such that for x∈Ωδ1g ( w ± ( d ( x ) ) ) = g ( η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) ) + η ± ψ ( ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 ) g ′ ( Θ ± ( d ( x ) ) ) ( C ± ∗ ± ε ) M ( d ( x ) ) . $$\begin{split}g(w_{\pm}(d(x)))&=g\bigg(\eta_{\pm}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)\bigg)\\&+\eta_{\pm}\psi\bigg(\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\bigg)g'(\Theta_{\pm}(d(x)))(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x)).\end{split}$$We obtain by Proposition 2.2 that (4.2) still holds. Moreover, we can adjust δ1 such that (4.3) holds here.Proof. The proof is still divided into the following two steps.Step 1. As before, for fixed ε> 0 and ∀ x∈Ωδ1, we define(4.29)r ( d ( x ) ) = ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N N + 1 $$\begin{equation}r(d(x))=\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N}{N+1}}\end{equation}$$andI 1 ± ( d ( x ) ) = ( M ( d ( x ) ) ) − 1 { [ ( N N + 1 ) N ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] m ^ ± − g ( ξ ± ψ ( r ( d ( x ) ) ) ) ξ ± N g ( ψ ( r ( d ( x ) ) ) ) } ; I 2 ± ( d ( x ) ) = C ± ∗ [ ( N N + 1 ) N N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) C ± ∏ i = 1 N − 1 ( 1 − d ( x ) κ i ( x ¯ ) ) − 1 − η ± ψ ( r ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) g ′ ( Θ ± ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ] ± ε [ ( N N + 1 ) N N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) m ^ + ∏ i = 1 N − 1 ( 1 − d ( x ) κ i ( x ¯ ) ) − 1 − η ± ψ ( r ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) g ′ ( Θ ± ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ] ; I 3 ± ( d ( x ) ) = − ( B 0 ± ε ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ ( M ( d ( x ) ) ) − 1 + ( M ( d ( x ) ) ) − 1 [ ( N N + 1 ) N ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] m ^ ± ∑ i = 1 N − 1 C N − 1 i ( − 1 ) i + 1 ( m ± d ( x ) ) i ( 1 − d ( x ) m ± ) N − 1 − ( B 0 ± ε ) ( C ± ∗ ± ε ) η ± ψ ( r ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) g ′ ( Θ ± ( d ( x ) ) ) g ′ ( η ± ψ ( r ( d ( x ) ) ) ) g ( η ± ψ ( r ( d ( x ) ) ) ) η ± N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ + ( N N + 1 ) N ζ ± ∗ ( d ( x ) ) ( M ( d ( x ) ) ) − 1 ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) , $$\begin{align}&I_{1\pm}(d(x))=(\mathfrak{M}(d(x)))^{-1}\bigg\{\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\ &+\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]\hat{m}_{\pm}-\frac{g(\xi_{\pm}\psi(r(d(x))))}{\xi_{\pm}^{N}g(\psi(r(d(x))))}\bigg\};\\ &I_{2\pm}(d(x))=C_{\pm}^{*}\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}N\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\mathscr{C}_{\pm}\prod\limits_{i=1}^{N-1}(1-d(x)\kappa_{i}(\bar{x}))^{-1}\\ &-\frac{\eta_{\pm}\psi(r(d(x)))g'(\eta_{\pm}\psi(r(d(x))))}{g(\eta_{\pm}\psi(r(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\eta_{\pm}\psi(r(d(x))))}\frac{g(\eta_{\pm}\psi(r(d(x))))}{\eta_{\pm}^{N}g(\psi(r(d(x))))}\bigg]\\ &\pm\varepsilon\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}N\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\hat{m}_{+}\prod\limits_{i=1}^{N-1}(1-d(x)\kappa_{i}(\bar{x}))^{-1}\\ &-\frac{\eta_{\pm}\psi(r(d(x)))g'(\eta_{\pm}\psi(r(d(x))))}{g(\eta_{\pm}\psi(r(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\eta_{\pm}\psi(r(d(x))))}\frac{g(\eta_{\pm}\psi(r(d(x))))}{\eta_{\pm}^{N}g(\psi(r(d(x))))}\bigg];\\ &I_{3\pm}(d(x))=-(B_{0}\pm\varepsilon)\frac{g(\eta_{\pm}\psi(r(d(x))))}{\eta_{\pm}^{N}g(\psi(r(d(x))))}(d(x))^{\mu}(\mathfrak{M}(d(x)))^{-1}\\ &+(\mathfrak{M}(d(x)))^{-1}\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\ &+\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]\frac{\hat{m}_{\pm}\sum_{i=1}^{N-1}C_{N-1}^{i}(-1)^{i+1}(m_{\pm}d(x))^{i}}{(1-d(x)m_{\pm})^{N-1}}\\ &-(B_{0}\pm\varepsilon)(C_{\pm}^{*}\pm\varepsilon)\frac{\eta_{\pm}\psi(r(d(x)))g'(\eta_{\pm}\psi(r(d(x))))}{g(\eta_{\pm}\psi(r(d(x))))}\frac{g'(\Theta_{\pm}(d(x)))}{g'(\eta_{\pm}\psi(r(d(x))))}\frac{g(\eta_{\pm}\psi(r(d(x))))}{\eta_{\pm}^{N}g(\psi(r(d(x))))}(d(x))^{\mu}\\ &+\bigg(\frac{N}{N+1}\bigg)^{N}\zeta_{\pm}^{*}(d(x))(\mathfrak{M}(d(x)))^{-1}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})},\end{align}$$whereif(S2) holds and θ=0,C±=mˆ±, if X+≥0 and X−≥0,C∓=mˆ∓, if X+≤0 and X−≤0,C+=C−=mˆ+, if X+>0 and X−<0,C+=C−=mˆ−, if X+<0 and X−>0; if (S1)holds,C±=mˆ±,$$\begin{cases}\mbox{if}\mathbf{(S_{2})} \mbox{ holds and }\theta=0,\,&\begin{cases} \mathscr{C}_{\pm}=\hat{m}_{\pm},\,&\mbox{ if }\mathfrak{X}_{+}\geq0 \mbox{ and }\mathfrak{X}_{-}\geq0,\\\mathscr{C}_{\mp}=\hat{m}_{\mp},\,&\mbox{ if }\mathfrak{X}_{+}\leq0 \mbox{ and }\mathfrak{X}_{-}\leq0,\\\mathscr{C}_{+}=\mathscr{C}_{-}=\hat{m}_{+},\,&\mbox{ if }\mathfrak{X}_{+}>0 \mbox{ and }\mathfrak{X}_{-}<0,\\\mathscr{C}_{+}=\mathscr{C}_{-}=\hat{m}_{-},\,&\mbox{ if }\mathfrak{X}_{+}<0 \mbox{ and }\mathfrak{X}_{-}>0;\end{cases}\\\mbox{ if }\mathbf{(S_{1})} \mbox{holds},\,\mathscr{C}_{\pm}=\hat{m}_{\pm},\end{cases}$$m ± are defined by (4.6) andζ ± ∗ = ( C ± ∗ ± ε ) ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) M ( d ( x ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( N + ( C ± ∗ ± ε ) ( N − 1 ) M ( d ( x ) ) ) + ( C ± ∗ ± ε ) 2 ( N − 1 ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ′ ( d ( x ) ) d ( x ) L ( d ( x ) ) − 1 ) × ( M ( d ( x ) ) ) 2 + [ β N N + 1 ( C ± ∗ ± ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( M ( d ( x ) ) ) β + 1 β + β N N + 1 ( C ± ± ε ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s ( M ( d ( x ) ) ) β + 1 β + β ( β + 1 ) N ( C ± ± ε ) N + 1 × ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s ( M ( d ( x ) ) ) β + 2 β + β ( C ± ± ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + L ~ ′ ( d ( x ) ) d ( x ) L ( d ( x ) ) − 1 ) ( M ( d ( x ) ) ) β + 1 β ] × ( 1 + ( C ± ∗ ± ε ) ( N − 1 ) M ( d ( x ) ) + ν ± ∗ ( d ( x ) ) ) + ν ± ∗ ( d ( x ) ) [ ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) ( 1 + ( C ± ∗ ± ε ) M ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ( 1 + ( C ± ± ε ) M ( d ( x ) ) ) ] $$ \begin{split} &\zeta_{\pm}^{*}=(C^{*}_{\pm}\pm\varepsilon)\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\mathfrak{M}(d(x))\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds} \\ &\times\big(N+(C^{*}_{\pm}\pm\varepsilon)(N-1)\mathfrak{M}(d(x))\big)+(C^{*}_{\pm}\pm\varepsilon)^{2}(N-1)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{L'(d(x))d(x)}{L(d(x))}-1\bigg) \\ &\times(\mathfrak{M}(d(x)))^{2}+\bigg[\frac{\beta N}{N+1}(C^{*}_{\pm}\pm\varepsilon)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds} \\ &\times(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}+\frac{\beta N}{N+1}(C_{\pm}\pm\varepsilon)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}+\frac{\beta(\beta+1)N(C_{\pm}\pm\varepsilon)}{N+1} \\ &\times\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}(\mathfrak{M}(d(x)))^{\frac{\beta+2}{\beta}} \\ &+\beta(C_{\pm}\pm\varepsilon)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\tilde{L}'(d(x))d(x)}{L(d(x))}-1\bigg)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\bigg] \\ &\times\big(1+(C_{\pm}^{*}\pm\varepsilon)(N-1)\mathfrak{M}(d(x))+\nu_{\pm}^{*}(d(x))\big) \\ &+\nu_{\pm}^{*}(d(x))\bigg[\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\big(1+(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x))\big)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds} \\ &+\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\big(1+(C_{\pm}\pm\varepsilon)\mathfrak{M}(d(x))\big)\bigg] \end{split} $$withν±∗(d(x))=R2±∗(d(x))+∑i=2N−1CN−1i(R1±∗(d(x))+R2±∗(d(x)))i,$$\nu_{\pm}^{*}(d(x))=\mathfrak{R}_{2\pm}^{*}(d(x))+\sum_{i=2}^{N-1}C_{N-1}^{i}\big(\mathfrak{R}_{1\pm}^{*}(d(x))+\mathfrak{R}_{2\pm}^{*}(d(x))\big)^{i},$$where CiN-1 is given in (4.7) andR1±∗(d(x))=(C±∗±ε)M(d(x))andR2±∗(d(x))=β(C±∗±ε)(M(d(x)))β+1β.$$\mathfrak{R}_{1\pm}^{*}(d(x))=(C_{\pm}^{*}\pm\varepsilon)\mathfrak{M}(d(x))\mbox{ and }\mathfrak{R}_{2\pm}^{*}(d(x))=\beta(C_{\pm}^{*}\pm\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}.$$Next, we prove(4.30)limd(x)→0(I1±(d(x))+I2±(d(x))+I3±(d(x)))=±ε(γ+Nmˆ+mˆ±)η±−(γ+N).$$\begin{equation}\lim_{d(x)\rightarrow0}(I_{1\pm}(d(x))+I_{2\pm}(d(x))+I_{3\pm}(d(x)))=\pm\varepsilon\bigg(\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{\pm}}\bigg)\eta_{\pm}^{-(\gamma+N)}.\end{equation}$$To prove (4.30), we calculate the limits of I1 ± (d(x)), I2 ± (d(x)) and I3 ± (d(x)) as d(x)→0.∙ First, by Lemma 3.2 (ii)−(iii), Lemma 3.6 (iv), (vi)−(vii) and (ix), we obtain(4.31)limd(x)→0I1±(d(x))=−NM+1N(N+1)σmˆ±(γ−1)2+σlnη±η±γN, if S2 and g3−g4 hold with θ=0,0, if S1 holds ing2.$$\begin{equation}\lim _{d(x) \rightarrow 0} I_{1 \pm}(d(x))=\left\{\begin{array}{ll}-\frac{\left(\frac{N}{M+1}\right)^{N}(N+1) \sigma \hat{m}_{\pm}}{(\gamma-1)^{2}}+\frac{\sigma \ln \eta_{\pm}}{\eta_{\pm}^{\gamma} N}, & \text { if }\left(\mathbf{S}_{2}\right) \text { and }\left(\mathbf{g}_{3}\right)-\left(\mathbf{g}_{4}\right) \text { hold with } \theta=0, \\ 0, & \text { if }\left(\mathbf{S}_{1}\right) \text { holds } \mathrm{in}\left(\mathbf{g}_{2}\right).\end{array}\right.\end{equation} $$∙ Second, by (4.2), Lemma 3.6 (iv) and the choices of η ± and C* ± in Theorem 1.3, we obtain(4.32)limd(x)→0I2±(d(x))=C±∗[(NN+1)NN(γ+N)γ−1C±+γη±−(γ+N)]±ε(γ+Nmˆ+mˆ±)η±−(γ+N).$$\begin{equation}\begin{split}&\lim_{d(x)\rightarrow0}I_{2\pm}(d(x))=C_{\pm}^{*}\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}\frac{N(\gamma+N)}{\gamma-1}\mathscr{C}_{\pm}+\gamma\eta_{\pm}^{-(\gamma+N)}\bigg]\pm\varepsilon\bigg(\gamma+N\frac{\hat{m}_{+}}{\hat{m}_{\pm}}\bigg)\eta_{\pm}^{-(\gamma+N)}.\end{split}\end{equation}$$∙ ,Third, we conclude by Lemma 3.5, Lemma 3.6 (iii)−(iv) and a straightforward calculation that(4.33)limd(x)→0I3±(d(x))=0.$$\begin{equation}\lim_{d(x)\rightarrow0}I_{3\pm}(d(x))=0.\end{equation}$$Combining (4.31)-(4.33), we obtain (4.30) holds. By (b1), (b3) and (4.30), we see that there exists a sufficiently small positive constant δεδ1 such that for any x∈Ωδε(d(x))−(N+1)L˜N(d(x))(1+(B0−ε)(d(x))μ)≤b(x)≤(d(x))−(N+1)L˜N(d(x))(1+(B0+ε)(d(x))μ)$$(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\big(1+(B_{0}-\varepsilon)(d(x))^{\mu}\big)\leqb(x)\leq(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\big(1+(B_{0}+\varepsilon)(d(x))^{\mu}\big)$$and (4.4)-(4.5) hold here. Step 2. Letu_ε(x)=−η+ψ(r(d(x)))(1+(C+∗+ε)M(d(x))),x∈Ωδε,$$\underline{u}_{\varepsilon}(x)=-\eta_{+}\psi(r(d(x)))\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big),\,x\in\Omega_{\delta_{\varepsilon}},$$where r(d(x)) is given by (4.29). Theng(−u_ε(x))=g(η+ψ(r(d(x))))+η+ψ(r(d(x)))g′(Θ+(d(x)))(C+∗+ε)M(d(x)),x∈Ωδε,$$g(-\underline{u}_{\varepsilon}(x))=g(\eta_{+}\psi(r(d(x))))+\eta_{+}\psi(r(d(x)))g'(\Theta_{+}(d(x)))(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x)),\,x\in\Omega_{\delta_{\varepsilon}},$$where Θ+(d(x)) is given by (4.28). By Lemma 4.2, we have for any x∈Ωδε(4.34)det ( D 2 u _ ε ( x ) ) − b ( x ) g ( − u _ ε ( x ) ) ≥ − η + N ( N N + 1 ) N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) × L ~ N ( d ( x ) ) ( 1 + ( C + ∗ + ε ) ( N − 1 ) M ( d ( x ) ) + ν + ∗ ( d ( x ) ) ) [ ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) + β N N + 1 ( C + ∗ + ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( M ( d ( x ) ) ) β + 1 β + β N N + 1 ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( β + 1 ) N ( C + ∗ + ε ) N + 1 × ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 2 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( C + ∗ + ε ) × ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 1 β ( L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] × ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) ( 1 + ( B 0 + ε ) ( d ( x ) ) μ ) × [ g ( η + ψ ( r ( d ( x ) ) ) ) + η + ( C + ∗ + ε ) g ′ ( Θ + ( d ( x ) ) ) ψ ( r ( d ( x ) ) ) M ( d ( x ) ) ] = − η + N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) M ( d ( x ) ) { ( M ( d ( x ) ) ) − 1 × [ ( ( N N + 1 ) N ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ) m ^ + − g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ] + ( C + ∗ + ε ) [ ( N N + 1 ) N N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) − η + ψ ( r ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) g ′ ( Θ + ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ] − ( B 0 + ε ) g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ ( M ( d ( x ) ) ) − 1 + ( M ( d ( x ) ) ) − 1 [ ( N N + 1 ) N ( N N + 1 − 1 N + 1 × ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ( N N + 1 ) N ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] × m ^ + ∑ i = 1 N − 1 C N − 1 i ( − 1 ) i + 1 ( m + d ( x ) ) i ( 1 − d ( x ) m + ) N − 1 − ( B 0 + ε ) ( C + ∗ + ε ) η + ψ ( r ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) g ′ ( Θ + ( d ( x ) ) ) g ′ ( η + ψ ( r ( d ( x ) ) ) ) g ( η + ψ ( r ( d ( x ) ) ) ) η + N g ( ψ ( r ( d ( x ) ) ) ) ( d ( x ) ) μ + ( N N + 1 ) N ζ + ∗ ( d ( x ) ) ( M ( d ( x ) ) ) − 1 ∏ i = 1 N − 1 κ i ( x ¯ ) 1 − d ( x ) κ i ( x ¯ ) } ≥ − η + N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) M ( d ( x ) ) ∑ i = 1 3 I i + ( d ( x ) ) > 0 , $$\begin{align} \nonumber&\mbox{det}(D^{2}\underline{u}_{\varepsilon}(x))-b(x)g(-\underline{u}_{\varepsilon}(x))\geq-\eta_{+}^{N}\bigg(\frac{N}{N+1}\bigg)^{N}(\psi'(r(d(x))))^{N-1}\psi''(r(d(x)))(d(x))^{-(N+1)}\\ \nonumber&\times\tilde{L}^{N}(d(x))\big(1+(C_{+}^{*}+\varepsilon)(N-1)\mathfrak{M}(d(x))+\nu_{+}^{*}(d(x))\big)\bigg[\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\\ \nonumber&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\\ \nonumber&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big)+\frac{\beta N}{N+1}(C_{+}^{*}+\varepsilon)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\ \nonumber&\times(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}+\frac{\beta N}{N+1}(C_{+}^{*}+\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\beta(\beta+1)N(C_{+}^{*}+\varepsilon)}{N+1}\\ \label{ffff} &\times\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}(\mathfrak{M}(d(x)))^{\frac{\beta+2}{\beta}}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\beta(C_{+}^{*}+\varepsilon)\\ \nonumber&\times\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\bigg(\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]\\ \nonumber&\times\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}-(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\big(1+(B_{0}+\varepsilon)(d(x))^{\mu}\big)\\ \nonumber&\times\big[g(\eta_{+}\psi(r(d(x))))+\eta_{+}(C_{+}^{*}+\varepsilon)g'(\Theta_{+}(d(x)))\psi(r(d(x)))\mathfrak{M}(d(x))\big]\\ \nonumber&=-\eta_{+}^{N}(\psi'(r(d(x))))^{N-1}\psi''(r(d(x)))(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\mathfrak{M}(d(x))\bigg\{(\mathfrak{M}(d(x)))^{-1}\\ \nonumber&\times\bigg[\bigg(\bigg(\frac{N}{N+1}\bigg)^{N}\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\ \nonumber&+\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg)\hat{m}_{+}-\frac{g(\eta_{+}\psi(r(d(x))))}{\eta_{+}^{N}g(\psi(r(d(x))))}\bigg]\\ \nonumber&+(C_{+}^{*}+\varepsilon)\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}N\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\\ \nonumber&-\frac{\eta_{+}\psi(r(d(x)))g'(\eta_{+}\psi(r(d(x))))}{g(\eta_{+}\psi(r(d(x))))}\frac{g'(\Theta_{+}(d(x)))}{g'(\eta_{+}\psi(r(d(x))))}\frac{g(\eta_{+}\psi(r(d(x))))}{\eta_{+}^{N}g(\psi(r(d(x))))}\bigg]\\ \nonumber&-(B_{0}+\varepsilon)\frac{g(\eta_{+}\psi(r(d(x))))}{\eta_{+}^{N}g(\psi(r(d(x))))}(d(x))^{\mu}(\mathfrak{M}(d(x)))^{-1}+(\mathfrak{M}(d(x)))^{-1}\bigg[\bigg(\frac{N}{N+1}\bigg)^{N}\bigg(\frac{N}{N+1}-\frac{1}{N+1}\\ \nonumber&\times\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\bigg(\frac{N}{N+1}\bigg)^{N}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]\\ \nonumber&\times\frac{\hat{m}_{+}\sum_{i=1}^{N-1}C_{N-1}^{i}(-1)^{i+1}(m_{+}d(x))^{i}}{(1-d(x)m_{+})^{N-1}}\\ \nonumber&-(B_{0}+\varepsilon)(C_{+}^{*}+\varepsilon)\frac{\eta_{+}\psi(r(d(x)))g'(\eta_{+}\psi(r(d(x))))}{g(\eta_{+}\psi(r(d(x))))}\frac{g'(\Theta_{+}(d(x)))}{g'(\eta_{+}\psi(r(d(x))))}\frac{g(\eta_{+}\psi(r(d(x))))}{\eta_{+}^{N}g(\psi(r(d(x))))}(d(x))^{\mu}\\ \nonumber&+\bigg(\frac{N}{N+1}\bigg)^{N}\zeta_{+}^{*}(d(x))(\mathfrak{M}(d(x)))^{-1}\prod\limits_{i=1}^{N-1}\frac{\kappa_{i}(\bar{x})}{1-d(x)\kappa_{i}(\bar{x})}\bigg\}\\ \nonumber&\geq-\eta_{+}^{N}(\psi'(r(d(x))))^{N-1}\psi''(r(d(x)))(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\mathfrak{M}(d(x))\sum\limits_{i=1}^{3}I_{i+}(d(x))>0, \end{align}$$i.e. uε is a subsolution of Eq. (1.1) in Ωδε. Moreover, it follows by Lemma 4.3 that for i=1, ... ,NS i ( D 2 u _ ε ( x ) ) = η + i ( N N + 1 ) i ( ψ ′ ( r ( d ( x ) ) ) ) i ( d ( x ) ) − i L ~ i ( d ( x ) ) ( ∫ 0 d ( x ) L ~ ( s ) s d s ) − i N + 1 × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) + β ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β ) i S i ( ϵ 1 , ⋅ ⋅ ⋅ , ϵ N − 1 ) − η + i ( N N + 1 ) i ( ψ ′ ( r ( d ( x ) ) ) ) i − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( i + 1 ) L ~ i ( d ( x ) ) ( ∫ 0 d ( x ) L ~ ( s ) s d s ) N − i N + 1 × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) + β ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β ) i − 1 [ ( N N + 1 − 1 N + 1 ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) × ( 1 + ( C + ∗ + ε ) M ( d ( x ) ) ) + β N N + 1 ( C + ∗ + ε ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s × ( M ( d ( x ) ) ) β + 1 β + β N N + 1 ( C + ∗ + ε ) ( M ( d ( x ) ) ) β + 1 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( β + 1 ) N ( C + ∗ + ε ) N + 1 × ψ ′ ( r ( d ( x ) ) ) r ( d ( x ) ) ψ ( r ( d ( x ) ) ) ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 2 β L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + β ( C + ∗ + ε ) × ψ ′ ( r ( d ( x ) ) ) ψ ″ ( r ( d ( x ) ) ) r ( d ( x ) ) ( M ( d ( x ) ) ) β + 1 β ( L ~ ( d ( x ) ) ∫ 0 d ( x ) L ~ ( s ) s d s + L ~ ′ ( d ( x ) ) d ( x ) L ~ ( d ( x ) ) − 1 ) ] S i − 1 ( ϵ 1 , ⋅ ⋅ ⋅ , ϵ N − 1 ) . $$\begin{split}&S_{i}(D^{2}\underline{u}_{\varepsilon}(x))=\eta_{+}^{i}\bigg(\frac{N}{N+1}\bigg)^{i}(\psi'(r(d(x))))^{i}(d(x))^{-i}\tilde{L}^{i}(d(x))\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{-\frac{i}{N+1}}\\&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))+\beta(C_{+}^{*}+\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\big)^{i}S_{i}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1})\\&-\eta_{+}^{i}\bigg(\frac{N}{N+1}\bigg)^{i}(\psi'(r(d(x))))^{i-1}\psi''(r(d(x)))(d(x))^{-(i+1)}\tilde{L}^{i}(d(x))\bigg(\int\limits_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds\bigg)^{\frac{N-i}{N+1}}\\&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))+\beta(C_{+}^{*}+\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\big)^{i-1}\bigg[\bigg(\frac{N}{N+1}-\frac{1}{N+1}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg)\\&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big)\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\bigg(\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\\&\times\big(1+(C_{+}^{*}+\varepsilon)\mathfrak{M}(d(x))\big)+\frac{\betaN}{N+1}(C_{+}^{*}+\varepsilon)\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}\\&\times(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}+\frac{\betaN}{N+1}(C_{+}^{*}+\varepsilon)(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\beta(\beta+1)N(C_{+}^{*}+\varepsilon)}{N+1}\\&\times\frac{\psi'(r(d(x)))r(d(x))}{\psi(r(d(x)))}\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}(\mathfrak{M}(d(x)))^{\frac{\beta+2}{\beta}}\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\beta(C_{+}^{*}+\varepsilon)\\&\times\frac{\psi'(r(d(x)))}{\psi''(r(d(x)))r(d(x))}(\mathfrak{M}(d(x)))^{\frac{\beta+1}{\beta}}\bigg(\frac{\tilde{L}(d(x))}{\int_{0}^{d(x)}\frac{\tilde{L}(s)}{s}ds}+\frac{\tilde{L}'(d(x))d(x)}{\tilde{L}(d(x))}-1\bigg)\bigg]S_{i-1}(\epsilon_{1},\cdot\cdot\cdot,\epsilon_{N-1}).\end{split}\quad\quad\quad\quad\quad\quad\quad\quad\quad$$This implies that we can adjust the above positive constant δε such that for any x∈ΩδεS i ( D 2 u _ ε ( x ) ) > 0 for i = 1 , ⋅ ⋅ ⋅ , N . $$S_{i}(D^{2}\underline{u}_{\varepsilon}(x))>0 \,\mbox{for}\,i=1,\cdot\cdot\cdot,N.$$We obtain by Lemma 4.4 that D2uε is positive definite in Ωδε.Letu‾ε(x)=−η−ψ(r(d(x)))(1+(C−∗−ε)M(d(x))),x∈Ωδε$$\overline{u}_{\varepsilon}(x)=-\eta_{-}\psi(r(d(x)))\big(1+(C_{-}^{*}-\varepsilon)\mathfrak{M}(d(x))\big),\,x\in\Omega_{\delta_{\varepsilon}}$$By the same calculation as (4.34), we obtaindet ( D 2 u ¯ ε ( x ) ) − b ( x ) g ( − u ¯ ε ( x ) ) ≤ − η − N ( ψ ′ ( r ( d ( x ) ) ) ) N − 1 ψ ″ ( r ( d ( x ) ) ) ( d ( x ) ) − ( N + 1 ) L ~ N ( d ( x ) ) M ( d ( x ) ) ∑ i = 1 3 I i − ( d ( x ) ) < 0 , $$\begin{split}&\mbox{det}(D^{2}\overline{u}_{\varepsilon}(x))-b(x)g(-\overline{u}_{\varepsilon}(x))\\&\leq-\eta_{-}^{N}(\psi'(r(d(x))))^{N-1}\psi''(r(d(x)))(d(x))^{-(N+1)}\tilde{L}^{N}(d(x))\mathfrak{M}(d(x))\sum\limits_{i=1}^{3}I_{i-}(d(x))<0,\end{split}$$i.e. uε is a supersolution of Eq. (1.1) in Ωδε.Let u be the unique strictly convex solution to problem (1.1). Through the same argument as Theorem 1.1, we see that there exists a large constant M > 0 such thatu_ε(x)−Md(x)≤u(x)≤u‾ε(x)+Md(x),x∈Ωδε.$$\underline{u}_{\varepsilon}(x)-Md(x)\leq u(x)\leq\overline{u}_{\varepsilon}(x)+Md(x),\,x\in\Omega_{\delta_{\varepsilon}}.$$Hence, we have for any x∈ ΩδεC+∗+ε+M(M(d(x)))−1d(x)η+ψ(r(d(x)))≥(−u(x)η+ψ(r(d(x)))−1)(M(d(x)))−1;C−∗−ε−M(M(d(x)))−1d(x)η−ψ(r(d(x)))≤(−u(x)η−ψ(r(d(x)))−1)(M(d(x)))−1.$$\begin{split}C_{+}^{*}+\varepsilon+\frac{M(\mathfrak{M}(d(x)))^{-1}d(x)}{\eta_{+}\psi(r(d(x)))}&\geq\bigg(\frac{-u(x)}{\eta_{+}\psi(r(d(x)))}-1\bigg)(\mathfrak{M}(d(x)))^{-1};\\C_{-}^{*}-\varepsilon-\frac{M(\mathfrak{M}(d(x)))^{-1}d(x)}{\eta_{-}\psi(r(d(x)))}&\leq\bigg(\frac{-u(x)}{\eta_{-}\psi(r(d(x)))}-1\bigg)(\mathfrak{M}(d(x)))^{-1}.\end{split}$$We conclude from (3.17)-(3.18), Proposition 2.5 (i)−(ii) thatC+∗+ε≥lim supd(x)→0(−u(x)η+ψ(r(d(x)))−1)(M(d(x)))−1;C−∗−ε≤lim infd(x)→0(−u(x)η−ψ(r(d(x)))−1)(M(d(x)))−1.$$\begin{split}C_{+}^{*}+\varepsilon&\geq\limsup_{d(x)\rightarrow0}\bigg(\frac{-u(x)}{\eta_{+}\psi(r(d(x)))}-1\bigg)(\mathfrak{M}(d(x)))^{-1};\\C_{-}^{*}-\varepsilon&\leq\liminf_{d(x)\rightarrow0}\bigg(\frac{-u(x)}{\eta_{-}\psi(r(d(x)))}-1\bigg)(\mathfrak{M}(d(x)))^{-1}.\end{split}$$Letting ε → 0 , the proof is finished.

Journal

Advances in Nonlinear Analysis – de Gruyter

Published: Jan 1, 2022

Keywords: Monge-Ampère equations; Strictly convex solution; Singular boundary value problem; The second boundary behavior; 35B40; 35J25; 35J60; 35J75; 35J96

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Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

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Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

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