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Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearity

Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic... 1Introduction and main resultIn this paper, we consider the following Schrödinger system with coupled quadratic nonlinearity(1.1)−ε2Δv+P(x)v=μvw,x∈RN,−ε2Δw+Q(x)w=μ2v2+γw2,x∈RN,v>0,w>0,v,w∈H1RN,$$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right),\end{array}\right. \end{equation}$$where ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive potentials.System (1.1) arises from the cubic nonlinear Schrödinger equation(1.2)i∂ϕ∂z+r∇2ϕ+χ|ϕ|2ϕ=0,$$\begin{equation}i \frac{\partial \phi}{\partial z}+r \nabla^{2} \phi+\chi|\phi|^{2} \phi=0, \end{equation}$$which appears in the nonlinear optic theory and can be used to describe the formation and propagation of optical solutions in Kerr-type materials [6, 19]. Here ϕ is a slowly varying envelope of electric field, the real-valued parameter r and χ represent the relative strength and sign of dispersion/ diffraction and nonlinearity respectively, and z is the propagation distance coordinate. The Laplacian operator ∇2 can either be ∂2∂τ2$\frac{\partial^{2}}{\partial \tau^{2}}$fortemporal solitons with τ is the normalized retarded time, or ∇2=∑i=1N∂2∂xi2$\nabla^{2}=\sum_{i=1}^{N} \frac{\partial^{2}}{\partial x_{i}^{2}}$, where x = (x1, · · · , xN) is in the direction orthogonal to z. Solitary wave solutions to (1.2) and its generations have been studied in [4, 18].Also, (1.1) appears in the study of standing waves for the following nonlinear system(1.3)i∂ϕ1∂t=−ε2Δϕ1+(P(x)+μ)ϕ1−μϕ1ϕ2,(x,t)∈RN×R+,i∂ϕ2∂t=−ε2Δϕ2+(Q(x)+μ)ϕ2−μ2ϕ12−γϕ22,(x,t)∈RN×R+,$$\begin{equation}\left\{\begin{array}{ll}i \frac{\partial \phi_{1}}{\partial t}=-\varepsilon^{2} \Delta \phi_{1}+(P(x)+\mu) \phi_{1}-\mu\left|\phi_{1}\right| \phi_{2}, & (x, t) \in \mathbb{R}^{N} \times \mathbb{R}^{+}, \\ i \frac{\partial \phi_{2}}{\partial t}=-\varepsilon^{2} \Delta \phi_{2}+(Q(x)+\mu) \phi_{2}-\frac{\mu}{2}\left|\phi_{1}\right|^{2}-\gamma\left|\phi_{2}\right|^{2}, & (x, t) \in \mathbb{R}^{N} \times \mathbb{R}^{+},\end{array}\right. \end{equation}$$with the form ϕ1(x,t)=v(x)eiμt,ϕ2(x,t)=w(x)eiμt,$\phi_{1}(x, t)=v(x) e^{i \mu t}, \phi_{2}(x, t)=w(x) e^{i \mu t},$where i is the imaginary unit and ε is the Planck constant. When ε = 1 and γ = 0, the existence of ground state solutions of (1.3) was proved in [27]. Besides, (1.1) is closely related to the general parabolic system with coupled nonlinearity and the nonlinear evolution equations. For this information, we can refer to [11, 16, 23, 25, 26] and references therein.By contrast with the coupled Schrödinger system(1.3) with χ(2) nonlinearities, the following χ2 nonlinear Schrödinger system(1.4)i∂ϕ1∂t=−ε2Δϕ1+V1(x)ϕ1−μ1ϕ12ϕ1−βϕ22ϕ1,(x,t)∈RN×R+,i∂ϕ2∂t=−ε2Δϕ2+V2(x)ϕ2−μ2ϕ22ϕ2−βϕ12ϕ2,(x,t)∈RN×R+$$\begin{equation}\left\{\begin{array}{ll}i \frac{\partial \phi_{1}}{\partial t}=-\varepsilon^{2} \Delta \phi_{1}+V_{1}(x) \phi_{1}-\mu_{1}\left|\phi_{1}\right|^{2} \phi_{1}-\beta\left|\phi_{2}\right|^{2} \phi_{1}, & (x, t) \in \mathbb{R}^{N} \times \mathbb{R}^{+}, \\ i \frac{\partial \phi_{2}}{\partial t}=-\varepsilon^{2} \Delta \phi_{2}+V_{2}(x) \phi_{2}-\mu_{2}\left|\phi_{2}\right|^{2} \phi_{2}-\beta\left|\phi_{1}\right|^{2} \phi_{2}, & (x, t) \in \mathbb{R}^{N} \times \mathbb{R}^{+}\end{array}\right. \end{equation}$$has been extensively investigated. There are many interesting results about (1.4) under various assumptions of V1(x) and V2(x), one can refer to [1, 2, 3, 4, 7, 8, 12, 13, 14, 15, 17, 21, 24] and their references therein.In recent decades, system(1.1) and its related problems have attracted a lot of attention. When ε = 1, (1.1) reduces to(1.5)−Δv+P(x)v=μvw,x∈RN,−Δw+Q(x)w=μ2v2+γw2,x∈RN,v>0,w>0,v,w∈H1RN.$$\begin{equation}\left\{\begin{array}{ll}-\Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right).\end{array}\right. \end{equation}$$Applying the finite dimensional reduction method, Wang and Zhou [22] constructed the infinitely many non-radial positive solutions of (1.5) if the potential functions P(x), Q(x) are radial and satisfy some algebraic decay at infinity. Also, if ε is small, for any positive integer k ≤ N + 1, Tang and Xie [20] proved that (1.1) has a k spikes solution concentrating at some strict local maximum of P(x) and Q(x) by using the finite dimensional reduction provided that |P(x)−P(y)|≤L1|x−y|θ1$|P(x)-P(y)| \leq L_{1}|x-y|^{\theta_{1}}$and |Q(x) − Q(y)| ≤ L2|x − y|θ2 for some positive constants L1, L2, θ1, θ2.Here we want to mention that, very recently, Luo, Peng and Yan [13] revisited the following Schrödinger equation(1.6)−ε2Δu+V(x)u=up−1,u∈H1RN$$\begin{equation}-\varepsilon^{2} \Delta u+V(x) u=u^{p-1}, u \in H^{1}\left(\mathbb{R}^{N}\right) \end{equation}$$with 2 < p < 2*. Under the condition that V(x) obtains its local minimum or local maximum x0 at a closed N − 1 dimensional hypersurface, they obtained the existence of a positive single peak solution for (1.6) concentrating at x0 if x0 is non-degenerate critical point of ΔV and also verified the local uniqueness of single peak solutions by using local Pohazaev type identity.Motivated by [13, 20, 22], we want to apply the finite-dimensional reduction to study the existence of positive single peak solutions for (1.1). Our purpose here is to prove that (1.1) has a single peak solution concentrating at some non-degenerate critical point of Δ(P + Q) on a closed N − 1 dimensional hypersurface.To state our results, throughout this paper, we assume that P(x), Q(x) obtain their local minimum or local maximum at a closed N − 1 dimensional hypersurface Γ. Without loss of generality, we suppose that P(x) = Q(x) = 1 if x ∈ Γ and more precisely, we assume that P(x), Q(x) satisfies the following conditions.(H1)There exist δ > 0 and a closed smooth hypersurface Γ such that if y ∈ Γ, P(y), Q(y) = 1 and P(y), Q(y) > 1 (or P(y), Q(y) < 1) for any y ∈ Wδ \ Γ, where Wδ := {x ∈ ℝN , dist(x, Γ) < δ}.(H2)The level set Γt = {x : P(x), Q(x) = t} is a closed smooth hypersurface for t ∈ [1, 1 +ϑ] (or t ∈ [1−ϑ, 1]) for some small ϑ > 0. Also, for any xt ∈ Γt and x0 ∈ Γ, there holds |νt −ν| ≤ C|xt −x0| and |ςi,t −ςi| ≤ C|xt −x0|, i = 1,2, ··· , N − 1. Here, throughout this paper, we denote by νt(ν) the outward unit normal vector of Γt(Γ) at xt(x0), while we use ςi,t(ςi) to denote the i-th principal tangential unit vector of Γt(Γ) at xt(x0).(H3)For any x ∈ Br(x0), it holds P(x) = Q(x), where x0 ∈ Γ and r is a small positive constant.Remark 1.1Let F(x)=∑i=1Nxi2ai2−1$F(x)=\sum_{i=1}^{N} \frac{x_{i}^{2}}{a_{i}^{2}}-1$with ai>0,ai≠aj(i≠j)$a_{i}>0, a_{i} \neq a_{j}(i \neq j)$and Γ=x∈RN:F(x)=0.$\Gamma=\left\{x \in \mathbb{R}^{N}: F(x)=0\right\}.$TakeP(x)=Q(x)=F2+1, in W0,$$\begin{equation}P(x)=Q(x)=F^{2}+1, \text { in } W_{0}, \end{equation}$$where W0=x∈RN:∑i=1Nxi2ai2−1≤δ0$W_{0}=\left\{x \in \mathbb{R}^{N}:\left|\sum_{i=1}^{N} \frac{x_{i}^{2}}{a_{i}^{2}}-1\right| \leq \delta_{0}\right\}$for some small fixed δ0 > 0. Then we can use the above conditions to the potentials P(x), Q(x).Let us point out that if Γ is a local minimum (or local maximum) set of P(x) and Q(x), then for any x ∈ Γ,P(x)=1,∇P(x)=0$$\begin{equation}P(x)=1, \nabla P(x)=0 \end{equation}$$andQ(x)=1,∇Q(x)=0.$$\begin{equation}Q(x)=1, \nabla Q(x)=0. \end{equation}$$This implies that for any tangential vector ς of Γ at x, one hasDζ∇P(x)=0,Dς∇Q(x)=0,∀x∈Γ,$$\begin{equation}\left(D_{\zeta} \nabla\right) P(x)=0,\left(D_{\varsigma} \nabla\right) Q(x)=0, \forall x \in \Gamma, \end{equation}$$where Dς denotes the directional derivative at the direction ς.Let U be the unique positive radial solution of the following problem(1.7)−Δu+u=u2,u>0, in RN,u(0)=maxx∈RNu(x),u(x)∈H1RN.$$\begin{equation}\left\{\begin{array}{ll}-\Delta u+u=u^{2}, & u>0, \text { in } \mathbb{R}^{N}, \\ u(0)=\max _{x \in \mathbb{R}^{N}} u(x), & u(x) \in H^{1}\left(\mathbb{R}^{N}\right) .\end{array}\right. \end{equation}$$It is well-known in [10] that U(x) is strictly decreasing and its s order derivative satisfiesDsU(x)e|x||x|1−N2≤C$$\begin{equation}\left|D^{s} U(x)\right| e^{|x|}|x|^{\frac{1-N}{2}} \leq C \end{equation}$$for |s| ≤ 1 and some constant C > 0.For xε close to x0, if we denoteVˉε,xε=PxεUPxεx−xεε,$$\begin{equation}\bar{V}_{\varepsilon, x_{\varepsilon}}=P\left(x_{\varepsilon}\right) U\left(\frac{\sqrt{P\left(x_{\varepsilon}\right)}\left(x-x_{\varepsilon}\right)}{\varepsilon}\right), \end{equation}$$then Vˉε,xE$\bar{V}_{\varepsilon, x_{\mathcal{E}}}$solves(1.8)−ε2Δv+Pxεv=v2,v>0,x∈RN,vxε=maxx∈RNv(x),v∈H1RN.$$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P\left(x_{\varepsilon}\right) v=v^{2}, & v>0, \quad x \in \mathbb{R}^{N}, \\ v\left(x_{\varepsilon}\right)=\max _{x \in \mathbb{R}^{N}} v(x), & v \in H^{1}\left(\mathbb{R}^{N}\right) .\end{array}\right. \end{equation}$$Also, writeWˉε,xε=QxεUQxEx−xEε,$$\begin{equation}\bar{W}_{\varepsilon, x_{\varepsilon}}=Q\left(x_{\varepsilon}\right) U\left(\frac{\sqrt{Q\left(x_{\mathcal{E}}\right)}\left(x-x_{\mathcal{E}}\right)}{\varepsilon}\right), \end{equation}$$which is the solution of(1.9)−ε2Δw+Qxεw=w2,w>0,x∈RN,wxE=maxx∈RNw(x),w∈H1RN.$$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta w+Q\left(x_{\varepsilon}\right) w=w^{2}, & w>0, \quad x \in \mathbb{R}^{N}, \\ w\left(x_{\mathcal{E}}\right)=\max _{x \in \mathbb{R}^{N}} w(x), & w \in H^{1}\left(\mathbb{R}^{N}\right) .\end{array}\right. \end{equation}$$Take Vε,xε,Wε,χε=αVˉε,χε,βWˉε,xε with α=1μ2(μ−γ)μ,β=1μ. Then Vε,Xε,Wε,xε$\left(V_{\varepsilon, x_{\varepsilon}}, W_{\varepsilon, \chi_{\varepsilon}}\right)=\left(\alpha \bar{V}_{\varepsilon, \chi_{\varepsilon}}, \beta \bar{W}_{\varepsilon, x_{\varepsilon}}\right) \text { with } \alpha=\frac{1}{\mu} \sqrt{\frac{2(\mu-\gamma)}{\mu}}, \beta=\frac{1}{\mu} . \text { Then }\left(V_{\varepsilon, X_{\varepsilon}}, W_{\varepsilon, x_{\varepsilon}}\right)$solves(1.10)−ε2Δv+Pxεv=μvw,x∈RN,−ε2Δw+Qxεw=μ2v2+γw2,x∈RN,v>0,w>0,v,w∈H1RN$$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P\left(x_{\varepsilon}\right) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q\left(x_{\varepsilon}\right) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right)\end{array}\right. \end{equation}$$since (H3) holds.For ε > 0 small, we will use (Vε,xε ,Wε,xε ) to construct the single peak solutions concentrating at x0. First we give the following definitions.Definition 1.2We say that (vε , wε) is a single peak solution of (1.1) concentrating at x0 if there exist xε∈RN$x_{\mathcal{\varepsilon}} \in \mathbb{R}^{N}$with |xε − x0| = o(1) such thatvε−Vε,Xε,wε−Wε,Xεε=oεN2,$$\begin{equation}\left\|\left(v_{\varepsilon}-V_{\varepsilon, X_{\varepsilon}}, w_{\varepsilon}-W_{\varepsilon, X_{\varepsilon}}\right)\right\|_{\varepsilon}=o\left(\varepsilon^{\frac{N}{2}}\right), \end{equation}$$where ∥(u,v)∥ε=∫RNε2|∇u|2+P(x)u2+ε2|∇v|2+Q(x)v2dx$\|(u, v)\|_{\varepsilon}=\int\limits_{\mathbb{R}^{N}}\left(\varepsilon^{2}|\nabla u|^{2}+P(x) u^{2}+\varepsilon^{2}|\nabla v|^{2}+Q(x) v^{2}\right) d x$Definition 1.3We say that a critical point x0 ∈ Γ of Δ(P + Q) on Γ is non-degenerate if it holds∂2Px0+Qx0∂v2≠0,det∂2ΔPx0∂ςi∂ςj1≤i,j≤N−1+∂2ΔQx0∂ςi∂ςj1≤i,j≤N−1≠0.$$\begin{equation}\frac{\partial^{2}\left(P\left(x_{0}\right)+Q\left(x_{0}\right)\right)}{\partial v^{2}} \neq 0, \quad \operatorname{det}\left(\left(\frac{\partial^{2} \Delta P\left(x_{0}\right)}{\partial \varsigma_{i} \partial \varsigma_{j}}\right)_{1 \leq i, j \leq N-1}+\left(\frac{\partial^{2} \Delta Q\left(x_{0}\right)}{\partial \varsigma_{i} \partial \varsigma_{j}}\right)_{1 \leq i, j \leq N-1}\right) \neq 0. \end{equation}$$The main result of this paper is the following.Theorem 1.4Assume that (H1)− (H3) hold. If x0 ∈ Γ is a non-degenerate critical point of Δ(P + Q), then there exists ε0 > 0 such that (1.1) has a single peak solution (vε , wε) concentrating at x0 provided ε ∈ (0, ε0].As in [13, 20, 22], we mainly apply the finite dimensional reduction method to prove our main result. Compared with [13],we have to overcome much difficulties in the reduction process which involves some technical and careful computations due to the χ(2) nonlinearity appears. Moreover, to our best knowledge, our result exhibits a new phenomenon for the coupled Schrödinger system with χ(2) nonlinearity.Remark 1.5Combining the ideas from [9, 13], where in [9] the coupled nonlinear Gross-Pitaevskii system was studied, we guess that the following conclusions may hold.(1) On the basis of Theorem 1.4, further we can prove the local uniqueness of single peak solutions by using local Pohazaev type identity.(2) Under the conditions of Theorem 1.4, if Δ(P + Q) has an isolated maximum or minimum point x0 ∈ Γ, then for any integer k > 0, (1.1) has a k-peaks solution whose peaks cluster at x0.The structure of this paper is organized as follows. In section 2, we do some preliminaries and then we carry out the finite dimensional reduction. We will prove our main result in section 3. In the sequel, for simplicity of notations we write ∫$\int$f to mean the Lebesgue integral of f (x) in ℝN.2the finite dimensional reductionIn this section, we mainly give some preliminaries and do the finite dimensional reduction. Hereafter, for any function K(x) > 0, we define the Sobolev spaceHε,K1=u∈H1RN:∫ε2|∇u|2+K(x)u2<∞,$$\begin{equation}H_{\varepsilon, K}^{1}=\left\{u \in H^{1}\left(\mathbb{R}^{N}\right): \int\left(\varepsilon^{2}|\nabla u|^{2}+K(x) u^{2}\right)<\infty\right\}, \end{equation}$$endowed with the standard norm∥u∥ε,K=∫ε2|∇u|2+K(x)u212,$$\begin{equation}\|u\|_{\varepsilon, K}=\left(\int\left(\varepsilon^{2}|\nabla u|^{2}+K(x) u^{2}\right)\right)^{\frac{1}{2}}, \end{equation}$$which is induced by the inner product〈u,v〉ε,K=∫ε2∇u∇v+K(x)uv.$$\begin{equation}\langle u, v\rangle_{\varepsilon, K}=\int\left(\varepsilon^{2} \nabla u \nabla v+K(x) u v\right) . \end{equation}$$Now we define H to be the product space Hε,P1×Hε,Q1$H_{\varepsilon, P}^{1} \times H_{\varepsilon, Q}^{1}$with the norm∥(u,v)∥ε2=∥u∥ε,P2+∥v∥ε,Q2$$\begin{equation}\|(u, v)\|_{\varepsilon}^{2}=\|u\|_{\varepsilon, P}^{2}+\|v\|_{\varepsilon, Q}^{2}\end{equation}$$and setEε,xε=(φ,ψ)∈H:(φ,ψ),∂Vε,xε∂xi,∂Wε,Xε∂xiε=0$$\begin{equation}E_{\varepsilon, x_{\varepsilon}}=\left\{(\varphi, \psi) \in H:\left\langle(\varphi, \psi),\left(\frac{\partial V_{\varepsilon, x_{\varepsilon}}}{\partial x_{i}}, \frac{\partial W_{\varepsilon, X_{\varepsilon}}}{\partial x_{i}}\right)\right\rangle_{\varepsilon}=0\right\} \end{equation}$$for i = 1, · · · , N, where(φ,ψ),∂Vε,Xε∂xi,∂Wε,Xε∂xiε=∫ε2∇φ∇∂Vε,xε∂xi+P(x)∂Vε,xε∂xiφ+ε2∇ψ∇∂Wε,xε∂xi+Q(x)∂Wε,Xε∂xiψ.$$\begin{equation}\begin{array}{l}\left\langle(\varphi, \psi),\left(\frac{\partial V_{\varepsilon, X_{\varepsilon}}}{\partial x_{i}}, \frac{\partial W_{\varepsilon, X_{\varepsilon}}}{\partial x_{i}}\right)\right\rangle_{\varepsilon} \\ =\int\left(\varepsilon^{2} \nabla \varphi \nabla \frac{\partial V_{\varepsilon, x_{\varepsilon}}}{\partial x_{i}}+P(x) \frac{\partial V_{\varepsilon, x_{\varepsilon}}}{\partial x_{i}} \varphi+\varepsilon^{2} \nabla \psi \nabla \frac{\partial W_{\varepsilon, x_{\varepsilon}}}{\partial x_{i}}+Q(x) \frac{\partial W_{\varepsilon, X_{\varepsilon}}}{\partial x_{i}} \psi\right) .\end{array} \end{equation}$$Note that the variational functional corresponding to (1.1) is(2.1)Iε(v,w)=12∫ε2|∇v|2+P(x)v2+ε2|∇w|2+Q(x)w2−μ2∫v2w−γ3∫w3.$$\begin{equation}I_{\varepsilon}(v, w)=\frac{1}{2} \int\left(\varepsilon^{2}|\nabla v|^{2}+P(x) v^{2}+\varepsilon^{2}|\nabla w|^{2}+Q(x) w^{2}\right)-\frac{\mu}{2} \int v^{2} w-\frac{\gamma}{3} \int w^{3} . \end{equation}$$Then I ∈ C2(H, ℝ) and its critical points are solutions of (1.1).SetJε(φ,ψ)=IεVε,xε+φ,Wε,xε+ψ,(φ,ψ)∈Eε,xε.$$\begin{equation}J_{\varepsilon}(\varphi, \psi)=I_{\varepsilon}\left(V_{\varepsilon, x_{\varepsilon}}+\varphi, W_{\varepsilon, x_{\varepsilon}}+\psi\right), \quad(\varphi, \psi) \in E_{\varepsilon, x_{\varepsilon}}. \end{equation}$$We can expand Jε(φ, ψ) as follows:(2.2)Jε(φ,ψ)=Jε(0,0)+ℓε(φ,ψ)+12Lε(φ,ψ)+Rε(φ,ψ),$$\begin{equation}J_{\varepsilon}(\varphi, \psi)=J_{\varepsilon}(0,0)+\ell_{\varepsilon}(\varphi, \psi)+\frac{1}{2} L_{\varepsilon}(\varphi, \psi)+R_{\varepsilon}(\varphi, \psi), \end{equation}$$whereℓε(φ,ψ)=∫ε2∇Vε,xε∇φ+P(x)Vε,xεφ+ε2∇Wε,xε∇ψ+Q(x)Wε,xεψ−μ∫Vε,xεWε,xεφ−μ2∫Vε,xε2ψ−γ∫Wε,xε2ψ=∫P(x)−PxεVε,Xεφ+Q(x)−QxεWε,Xεψ,Lε(φ,ψ)=∫ε2|∇φ|2+P(x)φ2+ε2|∇ψ|2+Q(x)ψ2−∫μWε,xεφ2+2γWε,xεψ2+2μVε,xεφψ$$\begin{equation}\begin{array}{c}\ell_{\varepsilon}(\varphi, \psi)=\int\left(\varepsilon^{2} \nabla V_{\varepsilon, x_{\varepsilon}} \nabla \varphi+P(x) V_{\varepsilon, x_{\varepsilon}} \varphi+\varepsilon^{2} \nabla W_{\varepsilon, x_{\varepsilon}} \nabla \psi+Q(x) W_{\varepsilon, x_{\varepsilon}} \psi\right)-\mu \int V_{\varepsilon, x_{\varepsilon}} W_{\varepsilon, x_{\varepsilon}} \varphi \\ -\frac{\mu}{2} \int V_{\varepsilon, x_{\varepsilon}}^{2} \psi-\gamma \int W_{\varepsilon, x_{\varepsilon}}^{2} \psi \\ =\int\left(P(x)-P\left(x_{\varepsilon}\right)\right) V_{\varepsilon, X_{\varepsilon}} \varphi+\left(Q(x)-Q\left(x_{\varepsilon}\right)\right) W_{\varepsilon, X_{\varepsilon}} \psi, \\ L_{\varepsilon}(\varphi, \psi)=\int\left(\varepsilon^{2}|\nabla \varphi|^{2}+P(x) \varphi^{2}+\varepsilon^{2}|\nabla \psi|^{2}+Q(x) \psi^{2}\right)-\int\left(\mu W_{\varepsilon, x_{\varepsilon}} \varphi^{2}+2 \gamma W_{\varepsilon, x_{\varepsilon}} \psi^{2}\right. \\ \left.\quad+2 \mu V_{\varepsilon, x_{\varepsilon}} \varphi \psi\right)\end{array} \end{equation}$$andRε(φ,ψ)=−μ2∫φ2ψ−γ3∫ψ3.$$\begin{equation}R_{\varepsilon}(\varphi, \psi)=-\frac{\mu}{2} \int \varphi^{2} \psi-\frac{\gamma}{3} \int \psi^{3} . \end{equation}$$It follows from [5] that Vε,Xε+φ,Wε,xε+ψ$\left(V_{\varepsilon, X_{\varepsilon}}+\varphi, W_{\varepsilon, x_{\varepsilon}}+\psi\right)$is a critical point of Iε(v, w) if and only if (φ, ψ) is a critical point of Jε(φ, ψ). In order to find a critical point for Jε(φ, ψ), we need to discuss each terms in the expansion (2.2).Lemma 2.1There exists C > 0 independent of ε such thatRεi(φ,ψ)≤Cε−N2∥(φ,ψ)∥ε3−i,i=0,1,2,$$\begin{equation}\left\|R_{\varepsilon}^{i}(\varphi, \psi)\right\| \leq C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}^{3-i}, i=0,1,2, \end{equation}$$where Rεi(φ,ψ)$R_{\varepsilon}^{i}(\varphi, \psi)$denotes the i-th derivative of Rε(φ, ψ).Proof. Recall thatRε(φ,ψ)=−μ2∫φ2ψ−γ3∫ψ3.$$\begin{equation}R_{\varepsilon}(\varphi, \psi)=-\frac{\mu}{2} \int \varphi^{2} \psi-\frac{\gamma}{3} \int \psi^{3} . \end{equation}$$By the direct computations, we getRε′(φ,ψ),(ξ,η)=−μ2∫φ2η+2φψξ−γ∫ψ2η$$\begin{equation}\left\langle R_{\varepsilon}^{\prime}(\varphi, \psi),(\xi, \eta)\right\rangle=-\frac{\mu}{2} \int\left(\varphi^{2} \eta+2 \varphi \psi \xi\right)-\gamma \int \psi^{2} \eta \end{equation}$$andRε′′(φ,ψ)(ξ,η),(g,h)=−μ∫(φgη+φhξ+ψgξ)−2γ∫ψhη.$$\begin{equation}\left\langle R_{\varepsilon}^{\prime \prime}(\varphi, \psi)(\xi, \eta),(g, h)\right\rangle=-\mu \int(\varphi g \eta+\varphi h \xi+\psi g \xi)-2 \gamma \int \psi h \eta. \end{equation}$$Observe that by letting uε(x) = u(εx), it is easy to check(2.3)∥u∥LpRN≤Cε1p−12N∥u∥ε$$\begin{equation}\|u\|_{L^{p}\left(\mathbb{R}^{N}\right)} \leq C \varepsilon^{\left(\frac{1}{p}-\frac{1}{2}\right) N}\|u\|_{\varepsilon} \end{equation}$$with ∥u∥ε=∫ε2|∇u|2+u212 for 2≤p≤2⋆$\|u\|_{\varepsilon}=\left(\int\left(\varepsilon^{2}|\nabla u|^{2}+u^{2}\right)\right)^{\frac{1}{2}} \text { for } 2 \leq p \leq 2^{\star}$and some C > 0. Then we haveRE(φ,ψ)≤C∫|φ|2|ψ|+∫|ψ|3≤C∫|φ|323∫|ψ|313+C∫|ψ|3≤Cε−N2∥φ∥ε2∥ψ∥ε+Cε−N2∥φ∥ε3≤Cε−N2∥(φ,ψ)∥ε3,$$\begin{equation}\begin{aligned}\left|R_{\mathcal{E}}(\varphi, \psi)\right| & \leq C\left(\int|\varphi|^{2}|\psi|+\int|\psi|^{3}\right) \\ & \leq C\left(\int|\varphi|^{3}\right)^{\frac{2}{3}}\left(\int|\psi|^{3}\right)^{\frac{1}{3}}+C \int|\psi|^{3} \\ & \leq C \varepsilon^{-\frac{N}{2}}\|\varphi\|_{\varepsilon}^{2}\|\psi\|_{\varepsilon}+C \varepsilon^{-\frac{N}{2}}\|\varphi\|_{\varepsilon}^{3} \\ & \leq C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}^{3}, \end{aligned} \end{equation}$$Rε′(φ,ψ),(ξ,η)≤C∫|φ|2|η|+|φ||ψ||ξ|+|ψ|2|η|≤C∫|φ|323∫|η|313+C∫|φ|313∫|ψ|313∫|ξ|313+C∫|ψ|323∫|η|313≤Cε−N2∥φ∥ε2∥η∥ε+∥φ∥ε∥ψ∥ε∥ξ∥ε+∥ψ∥ε2∥η∥ε≤Cε−N2∥(φ,ψ)∥ε2∥(ξ,η)∥ε$$\begin{equation}\begin{array}{l}\left|R_{\varepsilon}^{\prime}(\varphi, \psi),(\xi, \eta)\right| \\ \leq C \int\left(|\varphi|^{2}|\eta|+|\varphi||\psi||\xi|+|\psi|^{2}|\eta|\right) \\ \leq C\left(\int|\varphi|^{3}\right)^{\frac{2}{3}}\left(\int|\eta|^{3}\right)^{\frac{1}{3}}+C\left(\int|\varphi|^{3}\right)^{\frac{1}{3}}\left(\int|\psi|^{3}\right)^{\frac{1}{3}}\left(\int|\xi|^{3}\right)^{\frac{1}{3}}+C\left(\int|\psi|^{3}\right)^{\frac{2}{3}}\left(\int|\eta|^{3}\right)^{\frac{1}{3}} \\ \leq C \varepsilon^{-\frac{N}{2}}\left(\|\varphi\|_{\varepsilon}^{2}\|\eta\|_{\varepsilon}+\|\varphi\|_{\varepsilon}\|\psi\|_{\varepsilon}\|\xi\|_{\varepsilon}+\|\psi\|_{\varepsilon}^{2}\|\eta\|_{\varepsilon}\right) \\ \leq C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}^{2}\|(\xi, \eta)\|_{\varepsilon}\end{array} \end{equation}$$and similarly,Rε′′(φ,ψ)(ξ,η),(g,h)≤C∫(|φ||g||η|+|φ||h||ξ|+|ψ||g||ξ|+|ψ||h||η|)≤Cε−N2∥(φ,ψ)∥ε∥(ξ,η)∥ε∥(g,h)∥ε.$$\begin{equation}\begin{array}{l}\left|R_{\varepsilon}^{\prime \prime}(\varphi, \psi)(\xi, \eta),(g, h)\right| \\ \leq C \int(|\varphi||g||\eta|+|\varphi||h||\xi|+|\psi||g||\xi|+|\psi||h||\eta|) \\ \leq C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}\|(\xi, \eta)\|_{\varepsilon}\|(g, h)\|_{\varepsilon}.\end{array} \end{equation}$$This completes our proof.Lemma 2.2There exists C > 0 independent of ε such thatℓε≤C∇Pxε+∇QxεεN2+1+εN2+2.$$\begin{equation}\left\|\ell_{\varepsilon}\right\| \leq C\left(\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right). \end{equation}$$Proof. For any (φ,ψ)∈Eε,xε,$(\varphi, \psi) \in E_{\varepsilon, x_{\varepsilon}},$taking into account the decay property of U, we find for any fixed d > 0,ℓε(φ,ψ)=∫P(x)−PxεVε,xεφ+Q(x)−QxεWε,xεψ=εN∫Pεy+xε−PxεαPxεUPxεyφεy+xε+∫Qεy+xε−QxεβQxεUQxεyψεy+xε=εN∫Bdε(0)Pεy+xε−PxεαPxεUPxεyφεy+xε+∫Bdε(0)Qεy+xε−QxεβQxεUQxεyψεy+χε+∫RN∖Bdε(0)Pεy+xε−PxεαPxεUPxεyφεy+xε$$\begin{equation}\begin{aligned} \ell_{\varepsilon}(\varphi, \psi)=& \int\left(P(x)-P\left(x_{\varepsilon}\right)\right) V_{\varepsilon, x_{\varepsilon}} \varphi+\left(Q(x)-Q\left(x_{\varepsilon}\right)\right) W_{\varepsilon, x_{\varepsilon}} \psi \\=& \varepsilon^{N}\left(\int\left(P\left(\varepsilon y+x_{\varepsilon}\right)-P\left(x_{\varepsilon}\right)\right) \alpha P\left(x_{\varepsilon}\right) U\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right) \varphi\left(\varepsilon y+x_{\varepsilon}\right)\right.\\ &\left.\quad+\int\left(Q\left(\varepsilon y+x_{\varepsilon}\right)-Q\left(x_{\varepsilon}\right)\right) \beta Q\left(x_{\varepsilon}\right) U\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right) \psi\left(\varepsilon y+x_{\varepsilon}\right)\right) \\ =\varepsilon^{N} &\left\{\int_{B_{\frac{d}{\varepsilon}}(0)}\left(P\left(\varepsilon y+x_{\varepsilon}\right)-P\left(x_{\varepsilon}\right)\right) \alpha P\left(x_{\varepsilon}\right) U\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right) \varphi\left(\varepsilon y+x_{\varepsilon}\right)\right.\\ &+\int_{B_{\frac{d}{\varepsilon}}(0)}\left(Q\left(\varepsilon y+x_{\varepsilon}\right)-Q\left(x_{\varepsilon}\right)\right) \beta Q\left(x_{\varepsilon}\right) U\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right) \psi\left(\varepsilon y+\chi_{\varepsilon}\right) \\ &+\int_{\mathbb{R}^{N} \backslash B_{\frac{d}{\varepsilon}}(0)}\left(P\left(\varepsilon y+x_{\varepsilon}\right)-P\left(x_{\varepsilon}\right)\right) \alpha P\left(x_{\varepsilon}\right) U\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right) \varphi\left(\varepsilon y+x_{\varepsilon}\right) \end{aligned} \end{equation}$$+∫RN∖Bdε(0)Qεy+xε−QxεβQxεUQxεyψεy+xε≤CεN2∫Bdε(0)α2Pεy+xε−Pxε2P2xεU2Pxεy12+∫Bdε(0)β2Qεy+xε−Qxε2Q2xεU2Qxεy12∥(φ,ψ)∥ε+CεN2∬RN∖Bdε(0)U2Pxεy12+∫RN∖Bdε(0)U2Qxεy12∥(φ,ψ)∥ε=OεN2e−τε∥(φ,ψ)∥ε+OεN2∫dε∇Pxε|εy|+|εy|22U2Pxεy12+εN2∫Bdε(0)∇Qxε|εy|+|εy|22U2Qxεy12∥(φ,ψ)∥ε=O∇Pxε+∇QxεεN2+1+εN2+2∥(φ,ψ)∥ε$$\begin{equation}\begin{array}{c}\left.+\int_{\mathbb{R}^{N} \backslash B_{\frac{d}{\varepsilon}}(0)}\left(Q\left(\varepsilon y+x_{\varepsilon}\right)-Q\left(x_{\varepsilon}\right)\right) \beta Q\left(x_{\varepsilon}\right) U\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right) \psi\left(\varepsilon y+x_{\varepsilon}\right)\right\} \\ \leq C \varepsilon^{\frac{N}{2}}\left[\left(\int_{B_{\frac{d}{\varepsilon}}(0)} \alpha^{2}\left(P\left(\varepsilon y+x_{\varepsilon}\right)-P\left(x_{\varepsilon}\right)\right)^{2} P^{2}\left(x_{\varepsilon}\right) U^{2}\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right. \\ \left.+\left(\int_{B_{\frac{d}{\varepsilon}}(0)} \beta^{2}\left(Q\left(\varepsilon y+x_{\mathcal{\varepsilon}}\right)-Q\left(x_{\varepsilon}\right)\right)^{2} Q^{2}\left(x_{\varepsilon}\right) U^{2}\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right]\|(\varphi, \psi)\|_{\varepsilon} \\ +C \varepsilon^{\frac{N}{2}}\left[\left(\iint_{\mathbb{R}^{N} \backslash B_{\frac{d}{\varepsilon}}(0)} U^{2}\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}+\left(\int_{\mathbb{R}^{N} \backslash B_{\frac{d}{\varepsilon}}(0)} U^{2}\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right]\|(\varphi, \psi)\|_{\varepsilon} \\ =O\left(\varepsilon^{\frac{N}{2}} e^{-\frac{\tau}{\varepsilon}}\right)\|(\varphi, \psi)\|_{\varepsilon}+O\left[\varepsilon^{\frac{N}{2}}\left(\int_{\frac{d}{\varepsilon}}\left(\left|\nabla P\left(x_{\varepsilon}\right)\right||\varepsilon y|+|\varepsilon y|^{2}\right)^{2} U^{2}\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right. \\ \left.\quad+\varepsilon^{\frac{N}{2}}\left(\int_{B_{\frac{d}{\varepsilon}}(0)}\left(\left|\nabla Q\left(x_{\varepsilon}\right)\right||\varepsilon y|+|\varepsilon y|^{2}\right)^{2} U^{2}\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right]\|(\varphi, \psi)\|_{\varepsilon} \\ =O\left(\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right)\|(\varphi, \psi)\|_{\varepsilon}\end{array} \end{equation}$$for some τ > 0.Now it is easy to check that∫ε2(∇v∇φ+∇w∇ψ)+P(x)vφ+Q(x)wψ−∫μWε,xεvφ+μVε,xεvψ+μVε,xεwφ+2γWε,xεwψ$$\begin{equation}\begin{array}{c}\int\left(\varepsilon^{2}(\nabla v \nabla \varphi+\nabla w \nabla \psi)+P(x) v \varphi+Q(x) w \psi\right)-\int\left(\mu W_{\varepsilon, x_{\varepsilon}} v \varphi+\mu V_{\varepsilon, x_{\varepsilon}} v \psi+\mu V_{\varepsilon, x_{\varepsilon}} w \varphi\right. \\ \left.+2 \gamma W_{\varepsilon, x_{\varepsilon}} w \psi\right)\end{array} \end{equation}$$is a bounded bi-linear functional in Eε,xε.$E_{\varepsilon, x_{\varepsilon}}.$Hence we can define L to be a bounded linear map from Eε,xε$E_{\varepsilon, x_{\varepsilon}}$to Eε,xε such that for any (v,w),(φ,ψ)∈Eε,χε,$(v, w),(\varphi, \psi) \in E_{\varepsilon, \chi_{\varepsilon}},$〈L(v,w),(φ,ψ)〉=∫ε2(∇v∇φ+∇w∇ψ)+P(x)vφ+Q(x)wψ−∫μWε,xεvφ+μVε,xεvψ+μVε,xεwφ+2γWε,xεwψ.$$\begin{equation}\begin{aligned}\langle L(v, w),(\varphi, \psi)\rangle=\int &\left(\varepsilon^{2}(\nabla v \nabla \varphi+\nabla w \nabla \psi)+P(x) v \varphi+Q(x) w \psi\right) \\ &-\int\left(\mu W_{\varepsilon, x_{\varepsilon}} v \varphi+\mu V_{\varepsilon, x_{\varepsilon}} v \psi+\mu V_{\varepsilon, x_{\varepsilon}} w \varphi+2 \gamma W_{\varepsilon, x_{\varepsilon}} w \psi\right). \end{aligned} \end{equation}$$Next we want to discuss the invertibility of L in Eε,Xε.$E_{\varepsilon, X_{\varepsilon}}.$To this end, we denote (V,W) = (αU, βU)). Then (V,W) solves(2.4)−Δv+v=μvw,x∈RN,−Δw+w=μ2v2+γw2,x∈RN.$$\begin{equation}\left\{\begin{array}{ll}-\Delta v+v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\Delta w+w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}.\end{array}\right. \end{equation}$$It follows from Proposition 2.2 in [22] thatProposition 2.3For any μ > 0 and μ > γ, (V,W) is non-degenerate for the system (2.4) in H1RN×H1RN$H^{1}\left(\mathbb{R}^{N}\right) \times H^{1}\left(\mathbb{R}^{N}\right)$in the sence that the kernel is given by span λ(μ,γ)∂V∂xi,∂W∂xi∣i=1,2,⋯,N$\left\{\lambda(\mu, \gamma) \frac{\partial V}{\partial x_{i}}, \frac{\partial W}{\partial x_{i}} \mid i=1,2, \cdots, N\right\}$, where λ(μ,γ)≠0.$\lambda(\mu, \gamma) \neq 0.$Using the above result, we come to discuss the invertibility of L in Eε,Xε.$E_{\varepsilon, X_{\varepsilon}}.$Lemma 2.4There exist constants ρ > 0 and ε0 > 0 such that for all ε ∈ (0, ε0],∥L(v,w)∥≥ρ∥(v,w)∥ε,∀(v,w)∈Eε,xε.$$\begin{equation}\|L(v, w)\| \geq \rho\|(v, w)\|_{\varepsilon}, \quad \forall(v, w) \in E_{\varepsilon, x_{\varepsilon}}. \end{equation}$$Proof. We argue by contradiction. Suppose that there exist εn → 0, χεn$\chi_{\varepsilon_{n}}$→ x0 and (vn , wn) ∈ Eεn,xεn$E_{\varepsilon_{n}, x_{\varepsilon_{n}}}$such that for any φn,ψn∈Eεn,xεn,$\left(\varphi_{n}, \psi_{n}\right) \in E_{\varepsilon_{n}, x_{\varepsilon_{n}}},$(2.5)Lvn,wn,φn,ψn=on(1)vn,wnεnφn,ψnεn.$$\begin{equation}\left\langle L\left(v_{n}, w_{n}\right),\left(\varphi_{n}, \psi_{n}\right)\right\rangle=o_{n}(1)\left\|\left(v_{n}, w_{n}\right)\right\|_{\varepsilon_{n}}\left\|\left(\varphi_{n}, \psi_{n}\right)\right\|_{\varepsilon_{n}}. \end{equation}$$Without loss of generality, we can assume that vn,wnεn2=εnN$\left\|\left(v_{n}, w_{n}\right)\right\|_{\varepsilon_{n}}^{2}=\varepsilon_{n}^{N}$and letv˜n(x)=1PxεnvnεnχPxεn+χεn$$\begin{equation}\widetilde{v}_{n}(x)=\frac{1}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}} v_{n}\left(\frac{\varepsilon_{n} \chi}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+\chi_{\varepsilon_{n}}\right) \end{equation}$$andw˜n(x)=1QxεnwnεnxQxεn+χεn.$$\begin{equation}\widetilde{w}_{n}(x)=\frac{1}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}} w_{n}\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+\chi_{\varepsilon_{n}}\right). \end{equation}$$Then, in view of vn,wnεn2=εnN,$\left\|\left(v_{n}, w_{n}\right)\right\|_{\varepsilon_{n}}^{2}=\varepsilon_{n}^{N},$we getv˜n,w˜nH1×H1=∫∇v˜n2+v˜n2+∇w˜n2+w˜n2≤C,$$\begin{equation}\left\|\left(\widetilde{v}_{n}, \widetilde{w}_{n}\right)\right\|_{H^{1} \times H^{1}}=\int\left(\left|\nabla \widetilde{v}_{n}\right|^{2}+\widetilde{v}_{n}^{2}+\left|\nabla \widetilde{w}_{n}\right|^{2}+\widetilde{w}_{n}^{2}\right) \leq C, \end{equation}$$which implies that up to a subsequence, there exists v, w ∈ H1(ℝN) such that as n → +∞,v˜n⇀v,w˜n⇀w in H1RN,v˜n→v,w˜n→w in Lloc2RN.$$\begin{equation}\left\{\begin{array}{lll}\widetilde{v}_{n} \rightharpoonup v, & \widetilde{w}_{n} \rightharpoonup w & \text { in } H^{1}\left(\mathbb{R}^{N}\right), \\ \widetilde{v}_{n} \rightarrow v, & \widetilde{w}_{n} \rightarrow w & \text { in } L_{l o c}^{2}\left(\mathbb{R}^{N}\right).\end{array}\right. \end{equation}$$Now we claim that v = w = 0. Considering (2.5), we find(2.6)∫P2xεn∇v˜n∇φ˜n+∇w˜n∇ψ˜n−μ(βU)v˜nφ˜n−μ(αU)w˜nφ˜n−μ(αU)v˜nψ˜n−2γ(βU)w˜nψ˜n+PxεnPεnyPxεn+xεnv˜nφ˜n+QxεnQεnyQxεn+xεnw˜nψ˜n=PxεnN4on(1)∫P2xεn∇φ˜n2+Q2xεn∇ψ˜n2+PxεnPεnyPxεn+xεnφ˜n2+QxεnQεnyPxεn+xεnψ˜n212,$$\begin{equation}\begin{aligned} \int\left[P^{2}\left(x_{\varepsilon_{n}}\right)\right.&\left(\nabla \widetilde{v}_{n} \nabla \widetilde{\varphi}_{n}+\nabla \widetilde{w}_{n} \nabla \widetilde{\psi}_{n}-\mu(\beta U) \widetilde{v}_{n} \widetilde{\varphi}_{n}-\mu(\alpha U) \widetilde{w}_{n} \widetilde{\varphi}_{n}-\mu(\alpha U) \widetilde{v}_{n} \widetilde{\psi}_{n}\right.\\ &\left.-2 \gamma(\beta U) \widetilde{w}_{n} \widetilde{\psi}_{n}\right)+P\left(x_{\varepsilon_{n}}\right) P\left(\frac{\varepsilon_{n} y}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{v}_{n} \widetilde{\varphi}_{n} \\ &\left.+Q\left(x_{\varepsilon_{n}}\right) Q\left(\frac{\varepsilon_{n} y}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{w}_{n} \widetilde{\psi}_{n}\right] \\=\left(P\left(x_{\varepsilon_{n}}\right)\right)^{\frac{N}{4}} & o_{n}(1)\left[\int P^{2}\left(x_{\varepsilon_{n}}\right)\left|\nabla \widetilde{\varphi}_{n}\right|^{2}+Q^{2}\left(x_{\varepsilon_{n}}\right)\left|\nabla \widetilde{\psi}_{n}\right|^{2}+P\left(x_{\varepsilon_{n}}\right) P\left(\frac{\varepsilon_{n} y}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{\varphi}_{n}^{2}\right.\\ &\left.+Q\left(x_{\varepsilon_{n}}\right) Q\left(\frac{\varepsilon_{n} y}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{\psi}_{n}^{2}\right]^{\frac{1}{2}}, \end{aligned} \end{equation}$$whereφ˜n(x)=1PxεnφnεnXPxεn+xEn,ψ˜n(x)=1QxεnψnεnxQxεn+χεn$$\begin{equation}\begin{array}{l}\widetilde{\varphi}_{n}(x)=\frac{1}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}} \varphi_{n}\left(\frac{\varepsilon_{n} X}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right), \\ \widetilde{\psi}_{n}(x)=\frac{1}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}} \psi_{n}\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+\chi_{\varepsilon_{n}}\right)\end{array} \end{equation}$$and φ˜n,ψ˜n∈E˜εn,xεn$\left(\widetilde{\varphi}_{n}, \widetilde{\psi}_{n}\right) \in \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}$withE˜εn,xεn=(φ˜,ψ˜):φ˜Pxεnx−xEnεn,ψ˜Qxεnx−xεnεn∈Eεn,xεn.$$\begin{equation}\widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}=\left\{(\widetilde{\varphi}, \widetilde{\psi}):\left(\widetilde{\varphi}\left(\frac{\sqrt{P\left(x_{\varepsilon_{n}}\right)}\left(x-x_{\mathcal{E}_{n}}\right)}{\varepsilon_{n}}\right), \widetilde{\psi}\left(\frac{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}\left(x-x_{\varepsilon_{n}}\right)}{\varepsilon_{n}}\right)\right) \in E_{\varepsilon_{n}, x_{\varepsilon_{n}}}\right\}. \end{equation}$$On the other hand, being vn,wnεn2=εnN,$\left\|\left(v_{n}, w_{n}\right)\right\|_{\varepsilon_{n}}^{2}=\varepsilon_{n}^{N},$we have(2.7)∫Pxεn∇v˜n2+∇w˜n2+PεnχPxεn+xεnv˜n2+QεnxQxεn+xεnw˜n2=PxεnN2−1.$$\begin{equation}\begin{array}{c}\int\left[P\left(x_{\varepsilon_{n}}\right)\left(\left|\nabla \widetilde{v}_{n}\right|^{2}+\left|\nabla \widetilde{w}_{n}\right|^{2}\right)+P\left(\frac{\varepsilon_{n} \chi}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{v}_{n}^{2}\right. \\ \left.+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{w}_{n}^{2}\right]=\left(P\left(x_{\varepsilon_{n}}\right)\right)^{\frac{N}{2}-1}.\end{array} \end{equation}$$So, taking φn=vn,ψn=wn,$\varphi_{n}=v_{n}, \psi_{n}=w_{n},$(2.6) gives(2.8)∫Pxεn∇v˜n2+∇w˜n2−μ(βU)v˜n2−2μ(αU)v˜nw˜n−2γ(βU)w˜n2+PεnxPxεn+xEnv˜n2+QεnxQxεn+xEnw˜n2=on(1)PxεnN2−1.$$\begin{equation}\begin{array}{c}\int\left[P\left(x_{\varepsilon_{n}}\right)\left(\left|\nabla \widetilde{v}_{n}\right|^{2}+\left|\nabla \widetilde{w}_{n}\right|^{2}-\mu(\beta U) \widetilde{v}_{n}^{2}-2 \mu(\alpha U) \widetilde{v}_{n} \widetilde{w}_{n}-2 \gamma(\beta U)\left|\widetilde{w}_{n}\right|^{2}\right)\right. \\ \left.+P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right) \widetilde{v}_{n}^{2}+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right) \widetilde{w}_{n}^{2}\right] \\ =o_{n}(1)\left(P\left(x_{\varepsilon_{n}}\right)\right)^{\frac{N}{2}-1}.\end{array} \end{equation}$$Also, since v˜n,w˜n∈E˜εn,xεn$\left(\widetilde{v}_{n}, \widetilde{w}_{n}\right) \in \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}$, we obtain∫Pxεn∇v˜n∇∂(αU)∂yi+PεnxPxεn+xεnv˜n∂(αU)∂yi+Qxεn∇w˜n∇∂(βU)∂yi+QεnxQxεn+xεnw˜n∂(βU)∂yi=0,$$\begin{equation}\begin{array}{l}\int\left[P\left(x_{\varepsilon_{n}}\right) \nabla \widetilde{v}_{n} \nabla \frac{\partial(\alpha U)}{\partial y_{i}}+P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{v}_{n} \frac{\partial(\alpha U)}{\partial y_{i}}\right. \\ \left.\quad+Q\left(x_{\varepsilon_{n}}\right) \nabla \widetilde{w}_{n} \nabla \frac{\partial(\beta U)}{\partial y_{i}}+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{w}_{n} \frac{\partial(\beta U)}{\partial y_{i}}\right] \\ =0,\end{array} \end{equation}$$which gives that by letting n → +∞,(2.9)∫∇v∇∂(αU)∂xi+v(x)∂(αU)∂xi+∇w∇∂(βU)∂xi+w(x)∂(βU)∂xi=0.$$\begin{equation}\int\left[\nabla v \nabla \frac{\partial(\alpha U)}{\partial x_{i}}+v(x) \frac{\partial(\alpha U)}{\partial x_{i}}+\nabla w \nabla \frac{\partial(\beta U)}{\partial x_{i}}+w(x) \frac{\partial(\beta U)}{\partial x_{i}}\right]=0 . \end{equation}$$Now we take (φ˜,ψ˜)∈C0∞RN×C0∞RN$(\widetilde{\varphi}, \widetilde{\psi}) \in C_{0}^{\infty}\left(\mathbb{R}^{N}\right) \times C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$satisfying(2.10)∫∇φ˜∇∂(αU)∂xi+φ˜∂(αU)∂xi+∇ψ˜∇∂(βU)∂xi+ψ˜∂(βU)∂xi=0.$$\begin{equation}\int\left[\nabla \widetilde{\varphi} \nabla \frac{\partial(\alpha U)}{\partial x_{i}}+\widetilde{\varphi} \frac{\partial(\alpha U)}{\partial x_{i}}+\nabla \widetilde{\psi} \nabla \frac{\partial(\beta U)}{\partial x_{i}}+\widetilde{\psi} \frac{\partial(\beta U)}{\partial x_{i}}\right]=0. \end{equation}$$Meanwhile, we can decompose (φ˜,ψ˜)∈E˜εn,xεn$(\widetilde{\varphi}, \widetilde{\psi}) \in \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}$as follows(φ˜,ψ˜)=φ˜n,ψ˜n−∑i=1Nai,n∂Vεn,λεnεnXPxεn+xεn∂xi,∂Wεn,xεnεnXQxεn+xεn∂xi.$$\begin{equation}(\widetilde{\varphi}, \widetilde{\psi})=\left(\widetilde{\varphi}_{n}, \widetilde{\psi}_{n}\right)-\sum_{i=1}^{N} a_{i, n}\left(\frac{\partial V_{\varepsilon_{n}, \lambda_{\varepsilon_{n}}}\left(\frac{\varepsilon_{n} X}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right)}{\partial x_{i}}, \frac{\partial W_{\varepsilon_{n}, x_{\varepsilon_{n}}}\left(\frac{\varepsilon_{n} X}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right)}{\partial x_{i}}\right) . \end{equation}$$Then from φ˜n,ψ˜n∈E˜εn,xεn$\left(\widetilde{\varphi}_{n}, \widetilde{\psi}_{n}\right) \in \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}$and the definition ofE˜εn,xεn,$\widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}},$we get(2.11)∫Pxεn∇φ˜∇∂(αU)∂xi+PεnxPxεn+xεnφ˜∂(αU)∂xi+Qxεn∇ψ˜∇∂(βU)∂xi+QεnxQxεn+xεnψ˜∂(βU)∂xi+∫∑i=1Nai,n∇∂(αU)∂xi∇∂(αU)∂xi+PεnxPxεn+xεn∑i=1Nai,n∂(αU)∂xi∂(αU)∂xi+∑i=1Nai,n∇∂(βU)∂xi∇∂(βU)∂xi+QεnxQxεn+xεn∑i=1Nai,n∂(βU)∂xi∂(βU)∂xi=0.$$\begin{equation}\begin{array}{c}\int\left[P\left(x_{\varepsilon_{n}}\right) \nabla \widetilde{\varphi} \nabla \frac{\partial(\alpha U)}{\partial x_{i}}+P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{\varphi} \frac{\partial(\alpha U)}{\partial x_{i}}+Q\left(x_{\varepsilon_{n}}\right) \nabla \widetilde{\psi} \nabla \frac{\partial(\beta U)}{\partial x_{i}}\right. \\ \left.\quad+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{\psi} \frac{\partial(\beta U)}{\partial x_{i}}\right]+\int\left[\sum_{i=1}^{N} a_{i, n} \nabla \frac{\partial(\alpha U)}{\partial x_{i}} \nabla \frac{\partial(\alpha U)}{\partial x_{i}}\right. \\ +P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \sum_{i=1}^{N} a_{i, n} \frac{\partial(\alpha U)}{\partial x_{i}} \frac{\partial(\alpha U)}{\partial x_{i}}+\sum_{i=1}^{N} a_{i, n} \nabla \frac{\partial(\beta U)}{\partial x_{i}} \nabla \frac{\partial(\beta U)}{\partial x_{i}} \\ \left.+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \sum_{i=1}^{N} a_{i, n} \frac{\partial(\beta U)}{\partial x_{i}} \frac{\partial(\beta U)}{\partial x_{i}}\right]=0.\end{array} \end{equation}$$But by decay property of U, there exists C > 0 such that(2.12)∫∇∂(αU)∂xi2+∇∂(βU)∂xi2+∂(αU)∂xi2+∂(βU)∂xi2≥C>0.$$\begin{equation}\int\left[\left|\nabla \frac{\partial(\alpha U)}{\partial x_{i}}\right|^{2}+\left|\nabla \frac{\partial(\beta U)}{\partial x_{i}}\right|^{2}+\left|\frac{\partial(\alpha U)}{\partial x_{i}}\right|^{2}+\left|\frac{\partial(\beta U)}{\partial x_{i}}\right|^{2}\right] \geq C>0 . \end{equation}$$Hence, combining (2.10)-(2.12), we can easily check that ai,n → 0as n → +∞and then it follows from (2.6) that(2.13)∫[P2(x0)(∇v∇φ˜+∇w∇ψ˜+vφ˜+wψ˜−μ(βU)vφ˜−μ(αU)wφ˜−μ(αU)vψ˜−2γ(βU)wψ˜)]=0.$$\begin{aligned}\int \Big[P^2(x_0&)\big(\nabla v\nabla\widetilde{\varphi}+\nabla w\nabla\widetilde{\psi}+v\widetilde{\varphi}+w\widetilde{\psi} -\mu(\beta U)v\widetilde{\varphi}-\mu(\alpha U)w\widetilde{\varphi}\\ &-\mu(\alpha U)v\widetilde{\psi}-2\gamma(\beta U)w\widetilde{\psi}\big)\Big]=0. \end{aligned}$$This implies that for any (φ,ψ)∈C0∞RN×C0∞RN∩E˜εn,xεn,$(\varphi, \psi) \in C_{0}^{\infty}\left(\mathbb{R}^{N}\right) \times C_{0}^{\infty}\left(\mathbb{R}^{N}\right) \cap \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}},$(2.14)∫[∇v∇φ+∇w∇ψ+vφ+wψ−μ(βU)vφ−μ(αU)wφ−μ(αU)vψ−2γ(βU)wψ]=0.$$\begin{equation}\begin{array}{l}\int[\nabla v \nabla \varphi+\nabla w \nabla \psi+v \varphi+w \psi-\mu(\beta U) v \varphi-\mu(\alpha U) w \varphi \\ \quad-\mu(\alpha U) v \psi-2 \gamma(\beta U) w \psi]=0.\end{array} \end{equation}$$On the other hand, from the fact that (αU, βU) solves (2.4), we see(2.15)∫∇v∇∂(αU)∂xi+∇w∇∂(βU)∂xi+v∂(αU)∂xi+w∂(βU)∂xi−μ(βU)v∂(αU)∂xi−μ(αU)w∂(αU)∂xi−μ(αU)v∂(βU)∂xi−2γ(βU)w∂(βU)∂xi=0,$$\begin{equation}\begin{aligned} \int\left[\nabla v \nabla \frac{\partial(\alpha U)}{\partial x_{i}}\right.&+\nabla w \nabla \frac{\partial(\beta U)}{\partial x_{i}}+v \frac{\partial(\alpha U)}{\partial x_{i}}+w \frac{\partial(\beta U)}{\partial x_{i}}-\mu(\beta U) v \frac{\partial(\alpha U)}{\partial x_{i}}-\mu(\alpha U) w \frac{\partial(\alpha U)}{\partial x_{i}} \\\left.-\mu(\alpha U) v \frac{\partial(\beta U)}{\partial x_{i}}-2 \gamma(\beta U) w \frac{\partial(\beta U)}{\partial x_{i}}\right]=0, \end{aligned} \end{equation}$$which, together with (2.14), yields that for any (φ,ψ)∈C0∞RN×C0∞RN$(\varphi, \psi) \in C_{0}^{\infty}\left(\mathbb{R}^{N}\right) \times C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$∫[∇v∇φ+∇w∇ψ+vφ+wψ−μ(βU)vφ−μ(αU)wφ−μ(αU)vψ−2γ(βU)wψ]=0.$$\begin{equation}\begin{array}{l}\int[\nabla v \nabla \varphi+\nabla w \nabla \psi+v \varphi+w \psi-\mu(\beta U) v \varphi-\mu(\alpha U) w \varphi \\ \quad-\mu(\alpha U) v \psi-2 \gamma(\beta U) w \psi]=0.\end{array} \end{equation}$$So (v, w) is a solution of−Δv+v=μ(βU)v+μ(αU)w,x∈RN,−Δw+w=μ(αU)v+2γ(βU)w,x∈RN.$$\begin{equation}\left\{\begin{aligned}-\Delta v+v &=\mu(\beta U) v+\mu(\alpha U) w, & & x \in \mathbb{R}^{N}, \\-\Delta w+w &=\mu(\alpha U) v+2 \gamma(\beta U) w, & & x \in \mathbb{R}^{N} . \end{aligned}\right. \end{equation}$$Using Proposition 2.3, there exist bi ∈ ℝ, i = 1, 2, · · · , N such that(v,w)=∑i=1Nbi∂(αU)∂xi,∂(βU)∂xi.$$\begin{equation}(v, w)=\sum_{i=1}^{N} b_{i}\left(\frac{\partial(\alpha U)}{\partial x_{i}}, \frac{\partial(\beta U)}{\partial x_{i}}\right). \end{equation}$$But (2.9) gives that bi = 0, i = 1, 2, · · · , N. That is (v, w) = (0, 0), which is exactly our claim. Finally, taking into account that v˜n→0$\widetilde{v}_{n} \rightarrow 0$in Lloc2RN$L_{l o c}^{2}\left(\mathbb{R}^{N}\right)$and the exponential decay of U, we have∫Uv˜n2=∫BR(0)Uv˜n2+∫RN∣BR(0)Uv˜n2=oR(1)+Oe−R,$$\begin{equation}\int U\left|\widetilde{v}_{n}\right|^{2}=\int_{B_{R}(0)} U\left|\widetilde{v}_{n}\right|^{2}+\int_{\mathbb{R}^{N} \mid B_{R}(0)} U\left|\widetilde{v}_{n}\right|^{2}=o_{R}(1)+O\left(e^{-R}\right), \end{equation}$$where oR(1) → 0 as R → +∞.As a result, from (2.8) and (2.7), we deduce thaton(1)PxεnN2−1=∫Pxεn∇v˜n2+∇w˜n2−μ(βU)v˜n2−2μ(αU)v˜nw˜n−2γ(βU)w˜n2+PεnxPxεn+xEnv˜n2+QεnxQxεn+xEnw˜n2=PxEnN2−1+oR(1)+Oe−R,$$\begin{equation}\begin{array}{c}o_{n}(1)\left(P\left(x_{\varepsilon_{n}}\right)\right)^{\frac{N}{2}-1}=\int\left[P\left(x_{\varepsilon_{n}}\right)\left(\left|\nabla \widetilde{v}_{n}\right|^{2}+\left|\nabla \widetilde{w}_{n}\right|^{2}-\mu(\beta U) \widetilde{v}_{n}^{2}-2 \mu(\alpha U) \widetilde{v}_{n} \widetilde{w}_{n}-2 \gamma(\beta U)\left|\widetilde{w}_{n}\right|^{2}\right)\right. \\ \left.\quad+P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right) \widetilde{v}_{n}^{2}+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right) \widetilde{w}_{n}^{2}\right] \\ =\left(P\left(x_{\mathcal{E}_{n}}\right)\right)^{\frac{N}{2}-1}+o_{R}(1)+O\left(e^{-R}\right),\end{array} \end{equation}$$which is impossible for large n and R. So we complete this proof.Proposition 2.5For ε > 0 sufficiently small, there is (φε , ψε) ∈ Eε,xε such thatJε′φε,ψε,(g,h)=0,∀(g,h)∈Eε,xε.$$\begin{equation}\left\langle J_{\varepsilon}^{\prime}\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right),(g, h)\right\rangle=0, \quad \forall(g, h) \in E_{\varepsilon, x_{\varepsilon}}. \end{equation}$$Moreover,φε,ψεε≤C∇Pxε+∇QxεεN2+1+εN2+2$$\begin{equation}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon} \leq C\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right] \end{equation}$$for some constant C > 0 independent of ε.Proof. We will use the contraction mapping theorem to prove the wanted result. By Lemma 2.2, ℓε(φ,ψ)$\ell_{\varepsilon}(\varphi, \psi)$is a bounded linear function in Eε,Xε.$E_{\varepsilon, X_{\varepsilon}}.$So using Riesz representation theorem, we obtain that there is an ℓˉε∈Eε,xε,$\bar{\ell}_{\varepsilon} \in E_{\varepsilon, x_{\varepsilon}},$such thatℓε(φ,ψ)=ℓˉε,(φ,ψ).$$\begin{equation}\ell_{\varepsilon}(\varphi, \psi)=\left\langle\bar{\ell}_{\varepsilon},(\varphi, \psi)\right\rangle. \end{equation}$$Hence finding a critical point for Jε(φ, ψ) is equivalent to solving(2.16)ℓε+L(φ,ψ)+Rε′(φ,ψ)=0.$$\ell_{\varepsilon}+L(\varphi, \psi)+R_{\varepsilon}^{\prime}(\varphi, \psi)=0.$$It follows from Lemma 2.4 that (2.16) can be rewritten as(2.17)(φ,ψ)=A(φ,ψ):=−L−1ℓˉε+Rε′(φ,ψ).$$\begin{equation}(\varphi, \psi)=A(\varphi, \psi):=-L^{-1}\left(\bar{\ell}_{\varepsilon}+R_{\varepsilon}^{\prime}(\varphi, \psi)\right). \end{equation}$$Now we setSε=(φ,ψ)∈Eε,xε,∥(φ,ψ)∥ε≤∇Pxε+∇QxεεN2+1−θ+εN2+2−θ$$\begin{equation}S_{\varepsilon}=\left\{(\varphi, \psi) \in E_{\varepsilon, x_{\varepsilon}},\|(\varphi, \psi)\|_{\varepsilon} \leq\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1-\theta}+\varepsilon^{\frac{N}{2}+2-\theta}\right]\right\} \end{equation}$$with θ ∈ (0, 1). Then for any (φ, ψ) ∈ Sε,∥A(φ,ψ)∥≤Cℓˉεε+Rε′(φ,ψ)≤Cℓˉεε+Cε−N2∥(φ,ψ)∥ε2≤∇Pxε+∇QxεεN2+1−θ+εN2+2−θ.$$\begin{equation}\begin{aligned}\|A(\varphi, \psi)\| & \leq C\left(\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon}+\left\|R_{\varepsilon}^{\prime}(\varphi, \psi)\right\|\right) \\ & \leq C\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon}+C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}^{2} \\ & \leq\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1-\theta}+\varepsilon^{\frac{N}{2}+2-\theta}\right] . \end{aligned} \end{equation}$$Then A maps Sε to Sε. On the other hand, for any (φ1, ψ1), (φ2, ψ2) ∈ Sε, form Lemma 2.1,Aφ1,ψ1−Aφ2,ψ2=L−1Rε′φ1,ψ1−Rε′φ2,ψ2≤CRε′′tφ1,ψ1+(1−t)φ2,ψ2φ1,ψ1−φ2,ψ2ε≤12φ1,ψ1−φ2,ψ2ε,$$\begin{equation}\begin{aligned}\left\|A\left(\varphi_{1}, \psi_{1}\right)-A\left(\varphi_{2}, \psi_{2}\right)\right\| &=\left\|L^{-1}\left(R_{\varepsilon}^{\prime}\left(\varphi_{1}, \psi_{1}\right)-R_{\varepsilon}^{\prime}\left(\varphi_{2}, \psi_{2}\right)\right)\right\| \\ & \leq C\left\|R_{\varepsilon}^{\prime \prime}\left(t\left(\varphi_{1}, \psi_{1}\right)+(1-t)\left(\varphi_{2}, \psi_{2}\right)\right)\right\|\left\|\left(\varphi_{1}, \psi_{1}\right)-\left(\varphi_{2}, \psi_{2}\right)\right\|_{\varepsilon} \\ & \leq \frac{1}{2}\left\|\left(\varphi_{1}, \psi_{1}\right)-\left(\varphi_{2}, \psi_{2}\right)\right\|_{\varepsilon}, \end{aligned} \end{equation}$$where t ∈ (0, 1). This gives that A is a contraction map from Sε to Sε. Applying the contraction mapping theorem, we can find a unique (φε , ψε) ∈ Sε satisfying (2.17) andφε,ψεε≤∇Pxε+∇QxεεN2+1−θ+εN2+2−θ.$$\begin{equation}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon} \leq\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1-\theta}+\varepsilon^{\frac{N}{2}+2-\theta}\right] . \end{equation}$$Furthermore, in view of (2.17), we getφε,ψεε=L−1ℓˉε+L−1Rε′φε,ψεε≤Cℓˉεε+Cε−N2φε,ψεε2≤Cℓˉεε+C∇Pxε+∇QxEε1−θ+ε2−θφε,ψεε,$$\begin{equation}\begin{aligned}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon} &=\left\|L^{-1}\left(\bar{\ell}_{\varepsilon}\right)+L^{-1}\left(R_{\varepsilon}^{\prime}\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right)\right\|_{\varepsilon} \\ & \leq C\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon}+C \varepsilon^{-\frac{N}{2}}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon}^{2} \\ & \leq C\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon}+C\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\mathcal{E}}\right)\right|\right) \varepsilon^{1-\theta}+\varepsilon^{2-\theta}\right]\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon}, \end{aligned} \end{equation}$$which implies thatφε,ψεε≤Cℓˉεε≤C∇Pxε+∇QxεεN2+1+εN2+2.$$\begin{equation}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon} \leq C\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon} \leq C\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right]. \end{equation}$$3Proof of our main resultIn this section, we assume that x0 ∈ Γ is a non-degenerate critical point of Δ(P + Q) and we will construct a single peak solution (vε , wε) of (1.1) concentrating at x0.From Proposition 2.5, we can get the following result.Proposition 3.1There exists an ε0 > 0such that for any ε ∈ (0, ε0] and y close to x0, there is (φε,y , ψε,y) ∈ Eε,y such that for any (g, h) ∈ Eε,y,∫ε2∇Vε,y+φε,y∇g+P(x)Vε,y+φε,yg+ε2∇Wε,y+ψε,y∇h+Q(x)Wε,y+ψε,yh−∫μVε,y+φε,ygWε,y+ψε,y+μ2Vε,y+φε,y2h+γWε,y+ψε,y2h=0.$$\begin{equation}\begin{array}{l}\int\left[\varepsilon^{2} \nabla\left(V_{\varepsilon, y}+\varphi_{\varepsilon, y}\right) \nabla g+P(x)\left(V_{\varepsilon, y}+\varphi_{\varepsilon, y}\right) g+\varepsilon^{2} \nabla\left(W_{\varepsilon, y}+\psi_{\varepsilon, y}\right) \nabla h\right. \\ \left.+Q(x)\left(W_{\varepsilon, y}+\psi_{\varepsilon, y}\right) h\right]-\int\left[\mu\left(V_{\varepsilon, y}+\varphi_{\varepsilon, y}\right) g\left(W_{\varepsilon, y}+\psi_{\varepsilon, y}\right)+\frac{\mu}{2}\left(V_{\varepsilon, y}+\varphi_{\varepsilon, y}\right)^{2} h\right. \\ \left.+\gamma\left(W_{\varepsilon, y}+\psi_{\varepsilon, y}\right)^{2} h\right]=0.\end{array} \end{equation}$$Moreover,φε,y,ψε,yε=O(|∇P(y)|+|∇Q(y)|)εN2+1+εN2+2.$$\begin{equation}\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}=O\left((|\nabla P(y)|+|\nabla Q(y)|) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right) . \end{equation}$$To get a true solution of (1.1), we need to choose y such that(3.1)∫ε2∇vε∇∂vε∂xi+P(x)vε∂vε∂xi+ε2∇wε∇∂wε∂xi+Q(x)wε∂wε∂xi−∫μvε∂vε∂xiwε+μ2vε2∂wε∂xi+γwε2∂wε∂xi=0,$$\begin{equation}\begin{array}{c}\int\left[\varepsilon^{2} \nabla v_{\varepsilon} \nabla \frac{\partial v_{\varepsilon}}{\partial x_{i}}+P(x) v_{\varepsilon} \frac{\partial v_{\varepsilon}}{\partial x_{i}}+\varepsilon^{2} \nabla w_{\varepsilon} \nabla \frac{\partial w_{\varepsilon}}{\partial x_{i}}+Q(x) w_{\varepsilon} \frac{\partial w_{\varepsilon}}{\partial x_{i}}\right]-\int\left[\mu v_{\varepsilon} \frac{\partial v_{\varepsilon}}{\partial x_{i}} w_{\varepsilon}\right. \\ \left.+\frac{\mu}{2} v_{\varepsilon}^{2} \frac{\partial w_{\varepsilon}}{\partial x_{i}}+\gamma w_{\varepsilon}^{2} \frac{\partial w_{\varepsilon}}{\partial x_{i}}\right]=0,\end{array} \end{equation}$$where vε=Vε,y+φε,y,wε=Wε,y+ψε,y$v_{\varepsilon}=V_{\varepsilon, y}+\varphi_{\varepsilon, y}, w_{\varepsilon}=W_{\varepsilon, y}+\psi_{\varepsilon, y}$and i = 1, · · · , N. It is easy to check that (3.1) is equivalent to(3.2)∫∂P(x)∂xivε2+∂Q(x)∂xiwε2=0,i=1,⋯,N,$$\begin{equation}\int\left[\frac{\partial P(x)}{\partial x_{i}} v_{\varepsilon}^{2}+\frac{\partial Q(x)}{\partial x_{i}} w_{\varepsilon}^{2}\right]=0, i=1, \cdots, N, \end{equation}$$which is exact the Pohozaev type identity.For y close to x0, y ∈ Γt for some t close to 1. In the following, we denote by ν the unit normal vector of Γt at y and we use ςi , i = 1, · · · , N − 1, to denote the principal direction of Γt at y. Then, at y, one hasDζiP(y)=0,|∇P(y)|=DvP(y)$$\begin{equation}D_{\zeta_{i}} P(y)=0, \quad|\nabla P(y)|=\left|D_{v} P(y)\right| \end{equation}$$andDζiQ(y)=0,|∇Q(y)|=DvQ(y).$$\begin{equation}D_{\zeta_{i}} Q(y)=0,|\nabla Q(y)|=\left|D_{v} Q(y)\right|. \end{equation}$$First, we prove the following results.Lemma 3.2If (H1)− (H3) hold, then∫DvP(x)vε2+DvQ(x)wε2=0$$\begin{equation}\int\left(D_{v} P(x) v_{\varepsilon}^{2}+D_{v} Q(x) w_{\varepsilon}^{2}\right)=0 \end{equation}$$is equivalent toDvP(y)+DvQ(y)=Oε2.$$\begin{equation}D_{v} P(y)+D_{v} Q(y)=O\left(\varepsilon^{2}\right). \end{equation}$$Proof. By the direct computations, we have(3.3)∫DvP(x)Vε,y2+DvQ(x)Wε,y2=−∫2DvP(x)Vε,yφε,y+DvP(x)φε,y2+2DvQ(x)Wε,yψε,y+DvQ(x)ψε,y2=ODVP(y)+DvQ(y)εN2φε,y,ψε,yε+εN2+1φε,y,ψε,yε+φε,y,ψε,yε2=εNODvP(y)+DνQ(y)ε+ε2.$$\begin{equation}\begin{array}{l}\int\left(D_{v} P(x) V_{\varepsilon, y}^{2}+D_{v} Q(x) W_{\varepsilon, y}^{2}\right) \\ =-\int\left[2 D_{v} P(x) V_{\varepsilon, y} \varphi_{\varepsilon, y}+D_{v} P(x) \varphi_{\varepsilon, y}^{2}+2 D_{v} Q(x) W_{\varepsilon, y} \psi_{\varepsilon, y}+D_{v} Q(x) \psi_{\varepsilon, y}^{2}\right] \\ =O\left(\left(\left|D_{V} P(y)\right|+\left|D_{v} Q(y)\right|\right) \varepsilon^{\frac{N}{2}}\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}+\varepsilon^{\frac{N}{2}+1}\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}+\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}^{2}\right) \\ =\varepsilon^{N} O\left(\left(\left|D_{v} P(y)\right|+\left|D_{\nu} Q(y)\right|\right) \varepsilon+\varepsilon^{2}\right) .\end{array} \end{equation}$$On the other hand, using Taylor’s expansion, we get(3.4)∫DvP(x)Vε,y2+DvQ(x)Wε,y2=εN∫Bδε(0)DvP(y)+ε2|X|22!ΔDvP(y)α2P2(y)U2(P(y)x)+∫Bδε(0)DvQ(y)+ε2|X|22!ΔDvQ(y)β2Q2(y)U2(Q(y)x)+OεN+4=ODvP(y)+DvQ(y)εN+ΔDvP(y)+ΔDvQ(y)εN+2+εN+4.$$\begin{equation}\begin{array}{l}\int\left(D_{v} P(x) V_{\varepsilon, y}^{2}+D_{v} Q(x) W_{\varepsilon, y}^{2}\right) \\ =\varepsilon^{N}\left[\int_{B_{\frac{\delta}{\varepsilon}}(0)}\left(D_{v} P(y)+\frac{\varepsilon^{2}|X|^{2}}{2 !}\left(\Delta D_{v} P\right)(y)\right) \alpha^{2} P^{2}(y) U^{2}(\sqrt{P(y)} x)\right. \\ \left.+\int_{B_{\frac{\delta}{\varepsilon}}(0)}\left(D_{v} Q(y)+\frac{\varepsilon^{2}|X|^{2}}{2 !}\left(\Delta D_{v} Q\right)(y)\right) \beta^{2} Q^{2}(y) U^{2}(\sqrt{Q(y)} x)\right]+O\left(\varepsilon^{N+4}\right) \\ =O\left(\left(\left|D_{v} P(y)\right|+\left|D_{v} Q(y)\right|\right) \varepsilon^{N}+\left(\left(\Delta D_{v} P\right)(y)+\left(\Delta D_{v} Q\right)(y)\right) \varepsilon^{N+2}+\varepsilon^{N+4}\right).\end{array} \end{equation}$$Combining (3.3) and (3.4), we findDvP(y)+DvQ(y)=Oε2.$$\begin{equation}D_{v} P(y)+D_{v} Q(y)=O\left(\varepsilon^{2}\right). \end{equation}$$Lemma 3.3Under the conditions (H1)− (H3),∫DζP(x)vε2+DζQ(x)wε2=0$$\begin{equation}\int\left(D_{\zeta} P(x) v_{\varepsilon}^{2}+D_{\zeta} Q(x) w_{\varepsilon}^{2}\right)=0 \end{equation}$$is equivalent toDςΔP(y)+DςΔQ(y)+B1K(y)ε2=O∇G1(y)+∇G2(y)(|∇P(y)|+|∇Q(y)|)+∇G1(y)+∇G2(y)ε,$$\begin{equation}\begin{array}{l}\left(D_{\varsigma} \Delta P\right)(y)+\left(D_{\varsigma} \Delta Q\right)(y)+B_{1} K(y) \varepsilon^{2} \\ =O\left(\left(\left|\nabla G_{1}(y)\right|+\left|\nabla G_{2}(y)\right|\right)(|\nabla P(y)|+|\nabla Q(y)|)+\left(\left|\nabla G_{1}(y)\right|+\left|\nabla G_{2}(y)\right|\right) \varepsilon\right),\end{array} \end{equation}$$where B1 is some constant, K(y) is a smooth function, and G1(x) = ∇P(x), ς, G2(x) = ∇Q(x), ς.Proof. Since for any fixed d > 0 and j = 1,2Gj(x)=∑i=1N∂Gj(y)∂yixi−yi+12∑i=1N∑ℓ=1N∂2Gj(y)∂yi∂yℓxi−yixℓ−yℓ+o|x−y|2, in Bd(y),$$\begin{equation}G_{j}(x)=\sum_{i=1}^{N} \frac{\partial G_{j}(y)}{\partial y_{i}}\left(x_{i}-y_{i}\right)+\frac{1}{2} \sum_{i=1}^{N} \sum_{\ell=1}^{N} \frac{\partial^{2} G_{j}(y)}{\partial y_{i} \partial y_{\ell}}\left(x_{i}-y_{i}\right)\left(x_{\ell}-y_{\ell}\right)+o\left(|x-y|^{2}\right), \text { in } B_{d}(y), \end{equation}$$we have∫G1(x)Vε,y2+G2(x)Wε,y2=−2∫G1(x)Vε,yφε,y−∫G1(x)φε,y2−2∫G2(x)Wε,yψε,y−∫G2(x)ψε,y2=OεN2ε∇G1(y)+ε∇G2(y)+ε2φε,y,ψε,yε+ε∇G1(y)+∇G2(y)φε,y,ψε,yε2=O∇G1(y)+∇G2(y)(|∇P(y)|+|∇Q(y)|)εN+2+∇G1(y)+∇G2(y)εN+3+εN+4.$$\begin{equation}\begin{array}{l}\int\left(G_{1}(x) V_{\varepsilon, y}^{2}+G_{2}(x) W_{\varepsilon, y}^{2}\right) \\ =-2 \int G_{1}(x) V_{\varepsilon, y} \varphi_{\varepsilon, y}-\int G_{1}(x) \varphi_{\varepsilon, y}^{2}-2 \int G_{2}(x) W_{\varepsilon, y} \psi_{\varepsilon, y}-\int G_{2}(x) \psi_{\varepsilon, y}^{2} \\ =O\left(\varepsilon^{\frac{N}{2}}\left(\varepsilon\left|\nabla G_{1}(y)\right|+\varepsilon\left|\nabla G_{2}(y)\right|+\varepsilon^{2}\right)\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}+\varepsilon\left(\left|\nabla G_{1}(y)\right|\right.\right. \\ \left.\left.\quad+\left|\nabla G_{2}(y)\right|\right)\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}^{2}\right) \\ =O\left(\left(\left|\nabla G_{1}(y)\right|+\left|\nabla G_{2}(y)\right|\right)(|\nabla P(y)|+|\nabla Q(y)|) \varepsilon^{N+2}+\left(\left|\nabla G_{1}(y)\right|+\left|\nabla G_{2}(y)\right|\right) \varepsilon^{N+3}\right. \\ \left.\quad+\varepsilon^{N+4}\right).\end{array} \end{equation}$$On the other hand, from Gj(y) = 0, j = 1,2, we get∫G1(x)Vε,y2+G2(x)Wε,y2=OεN+2ΔG1(y)+ΔG2(y)+B1K(y)εN+4+εN+6.$$\begin{equation}\begin{array}{l}\int\left(G_{1}(x) V_{\varepsilon, y}^{2}+G_{2}(x) W_{\varepsilon, y}^{2}\right) \\ =O\left(\varepsilon^{N+2}\left(\Delta G_{1}(y)+\Delta G_{2}(y)\right)+B_{1} K(y) \varepsilon^{N+4}+\varepsilon^{N+6}\right).\end{array} \end{equation}$$As a result, the result follows.Proof of Theorem 1.4Now, by Lemmas 3.2 and 3.3, (3.2) is equivalent to(3.5)DvP(y)+DvQ(y)=Oε2,DςΔP(y)+DςΔQ(y)=O|∇P(y)|+|∇Q(y)|+ε2,$$\begin{equation}D_{v} P(y)+D_{v} Q(y)=O\left(\varepsilon^{2}\right),\left(D_{\varsigma} \Delta P\right)(y)+\left(D_{\varsigma} \Delta Q\right)(y)=O\left(|\nabla P(y)|+|\nabla Q(y)|+\varepsilon^{2}\right), \end{equation}$$which is also equivalent to(3.6)DvP(y)+DvQ(y)=Oε2,DςΔP(y)+DςΔQ(y)=Oε2.$$\begin{equation}D_{v} P(y)+D_{v} Q(y)=O\left(\varepsilon^{2}\right),\left(D_{\varsigma} \Delta P\right)(y)+\left(D_{\varsigma} \Delta Q\right)(y)=O\left(\varepsilon^{2}\right) . \end{equation}$$Let yˉ$\bar{y}$∈ Γ be the point such that y−yˉ=κv$y-\bar{y}=\kappa v$for some κ ∈ ℝ. We have DvP(yˉ)=0 and DvQ(yˉ)=0.$D_{v} P(\bar{y})=0 \text { and } D_{v} Q(\bar{y})=0.$As a result,DvP(y)+DvQ(y)=DvP(y)−DvP(yˉ)+DvQ(y)−DvQ(yˉ)=Dvv2P(yˉ)+Dvv2Q(yˉ)〈y−yˉ,v〉+O|y−yˉ|2,$$\begin{equation}\begin{aligned} D_{v} P(y)+D_{v} Q(y) &=D_{v} P(y)-D_{v} P(\bar{y})+D_{v} Q(y)-D_{v} Q(\bar{y}) \\ &=\left(D_{v v}^{2} P(\bar{y})+D_{v v}^{2} Q(\bar{y})\right)\langle y-\bar{y}, v\rangle+O\left(|y-\bar{y}|^{2}\right), \end{aligned} \end{equation}$$which, together with the non-degenerate assumption, yields that DνP(y) + DνQ(y) = O(ε2) can be written as(3.7)〈y−yˉ,v〉=Oε2+|y−yˉ|2.$$\begin{equation}\langle y-\bar{y}, v\rangle=O\left(\varepsilon^{2}+|y-\bar{y}|^{2}\right). \end{equation}$$Let ςˉi$\bar{\varsigma}_i$be the i − th tangential unit vector of Γ at yˉ.$\bar{y}.$It follows from the assumption (H2) that(DςiΔP)(y)+(DςiΔQ)(y)=(DςˉiΔP)(yˉ)+(DςˉiΔQ)(yˉ)+O(|y−yˉ|)=(DςˉiΔP)(yˉ)+(DςˉiΔQ)(yˉ)+O(ε2)$$\begin{aligned}(D_{\varsigma_i} \Delta P)(y)+(D_{\varsigma_i} \Delta Q)(y)&= (D_{\bar{\varsigma}_i} \Delta P)(\bar{y})+(D_{\bar{\varsigma}_i} \Delta Q)(\bar{y})+O(|y-\bar{y}|)\\ &=(D_{\bar{\varsigma}_i} \Delta P)(\bar{y})+(D_{\bar{\varsigma}_i} \Delta Q)(\bar{y})+O(\varepsilon^2) \end{aligned}$$and(DςˉiΔP)(yˉ)+(DςˉiΔQ)(yˉ)=(DςˉiΔP)(yˉ)−(Dςˉi,0ΔP)(x0)+(DςˉiΔQ)(yˉ)−(Dςˉi,0ΔQ)(x0)=〈(∇TDςi,0ΔP)(x0),yˉ−x0〉+〈(∇TDςi,0ΔQ)(x0),yˉ−x0〉+O(|yˉ−x0|2),$$\begin{aligned}&(D_{\bar{\varsigma}_i} \Delta P)(\bar{y})+(D_{\bar{\varsigma}_i} \Delta Q)(\bar{y})\\ &=(D_{\bar{\varsigma}_i} \Delta P)(\bar{y})-(D_{\bar{\varsigma}_{i,0}} \Delta P)(x_0)+ (D_{\bar{\varsigma}_i} \Delta Q)(\bar{y})-(D_{\bar{\varsigma}_{i,0}} \Delta Q)(x_0)\\ &=\langle (\nabla_T D_{\varsigma_{i,0}}\Delta P)(x_0),\bar{y}-x_0\rangle+\langle (\nabla_T D_{\varsigma_{i,0}}\Delta Q)(x_0),\bar{y}-x_0\rangle +O(|\bar{y}-x_0|^2), \end{aligned}$$where ∇T is the tangential gradient on Γ at x0 and ςi,0 is the i − th tangential unit vector of Γ at x0. Hence (DςΔP)(y) + (DςΔQ)(y) = O(ε2) can be rewritten as(3.8)〈(∇TDςi,0ΔP)(x0)+(∇TDςi,0ΔQ)(x0),yˉ−x0〉=O(ε2+|yˉ−x0|2).$$\begin{aligned}\langle (\nabla_T D_{\varsigma_{i,0}}\Delta P)(x_0)+ (\nabla_T D_{\varsigma_{i,0}}\Delta Q)(x_0),\bar{y}-x_0\rangle =O(\varepsilon^2+|\bar{y}-x_0|^2). \end{aligned}$$So we can solve (3.7) and (3.8) to get y = xε with xε → x0 as ε → 0. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Nonlinear Analysis de Gruyter

Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearity

Yang, Jing; Zhou, Ting
Advances in Nonlinear Analysis , Volume 11 (1): 15 – Jan 1, 2022
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de Gruyter
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© 2021 Jing Yang and Ting Zhou, published by De Gruyter
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2191-9496
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2191-950X
DOI
10.1515/anona-2021-0201
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Abstract

1Introduction and main resultIn this paper, we consider the following Schrödinger system with coupled quadratic nonlinearity(1.1)−ε2Δv+P(x)v=μvw,x∈RN,−ε2Δw+Q(x)w=μ2v2+γw2,x∈RN,v>0,w>0,v,w∈H1RN,$$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right),\end{array}\right. \end{equation}$$where ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive potentials.System (1.1) arises from the cubic nonlinear Schrödinger equation(1.2)i∂ϕ∂z+r∇2ϕ+χ|ϕ|2ϕ=0,$$\begin{equation}i \frac{\partial \phi}{\partial z}+r \nabla^{2} \phi+\chi|\phi|^{2} \phi=0, \end{equation}$$which appears in the nonlinear optic theory and can be used to describe the formation and propagation of optical solutions in Kerr-type materials [6, 19]. Here ϕ is a slowly varying envelope of electric field, the real-valued parameter r and χ represent the relative strength and sign of dispersion/ diffraction and nonlinearity respectively, and z is the propagation distance coordinate. The Laplacian operator ∇2 can either be ∂2∂τ2$\frac{\partial^{2}}{\partial \tau^{2}}$fortemporal solitons with τ is the normalized retarded time, or ∇2=∑i=1N∂2∂xi2$\nabla^{2}=\sum_{i=1}^{N} \frac{\partial^{2}}{\partial x_{i}^{2}}$, where x = (x1, · · · , xN) is in the direction orthogonal to z. Solitary wave solutions to (1.2) and its generations have been studied in [4, 18].Also, (1.1) appears in the study of standing waves for the following nonlinear system(1.3)i∂ϕ1∂t=−ε2Δϕ1+(P(x)+μ)ϕ1−μϕ1ϕ2,(x,t)∈RN×R+,i∂ϕ2∂t=−ε2Δϕ2+(Q(x)+μ)ϕ2−μ2ϕ12−γϕ22,(x,t)∈RN×R+,$$\begin{equation}\left\{\begin{array}{ll}i \frac{\partial \phi_{1}}{\partial t}=-\varepsilon^{2} \Delta \phi_{1}+(P(x)+\mu) \phi_{1}-\mu\left|\phi_{1}\right| \phi_{2}, & (x, t) \in \mathbb{R}^{N} \times \mathbb{R}^{+}, \\ i \frac{\partial \phi_{2}}{\partial t}=-\varepsilon^{2} \Delta \phi_{2}+(Q(x)+\mu) \phi_{2}-\frac{\mu}{2}\left|\phi_{1}\right|^{2}-\gamma\left|\phi_{2}\right|^{2}, & (x, t) \in \mathbb{R}^{N} \times \mathbb{R}^{+},\end{array}\right. \end{equation}$$with the form ϕ1(x,t)=v(x)eiμt,ϕ2(x,t)=w(x)eiμt,$\phi_{1}(x, t)=v(x) e^{i \mu t}, \phi_{2}(x, t)=w(x) e^{i \mu t},$where i is the imaginary unit and ε is the Planck constant. When ε = 1 and γ = 0, the existence of ground state solutions of (1.3) was proved in [27]. Besides, (1.1) is closely related to the general parabolic system with coupled nonlinearity and the nonlinear evolution equations. For this information, we can refer to [11, 16, 23, 25, 26] and references therein.By contrast with the coupled Schrödinger system(1.3) with χ(2) nonlinearities, the following χ2 nonlinear Schrödinger system(1.4)i∂ϕ1∂t=−ε2Δϕ1+V1(x)ϕ1−μ1ϕ12ϕ1−βϕ22ϕ1,(x,t)∈RN×R+,i∂ϕ2∂t=−ε2Δϕ2+V2(x)ϕ2−μ2ϕ22ϕ2−βϕ12ϕ2,(x,t)∈RN×R+$$\begin{equation}\left\{\begin{array}{ll}i \frac{\partial \phi_{1}}{\partial t}=-\varepsilon^{2} \Delta \phi_{1}+V_{1}(x) \phi_{1}-\mu_{1}\left|\phi_{1}\right|^{2} \phi_{1}-\beta\left|\phi_{2}\right|^{2} \phi_{1}, & (x, t) \in \mathbb{R}^{N} \times \mathbb{R}^{+}, \\ i \frac{\partial \phi_{2}}{\partial t}=-\varepsilon^{2} \Delta \phi_{2}+V_{2}(x) \phi_{2}-\mu_{2}\left|\phi_{2}\right|^{2} \phi_{2}-\beta\left|\phi_{1}\right|^{2} \phi_{2}, & (x, t) \in \mathbb{R}^{N} \times \mathbb{R}^{+}\end{array}\right. \end{equation}$$has been extensively investigated. There are many interesting results about (1.4) under various assumptions of V1(x) and V2(x), one can refer to [1, 2, 3, 4, 7, 8, 12, 13, 14, 15, 17, 21, 24] and their references therein.In recent decades, system(1.1) and its related problems have attracted a lot of attention. When ε = 1, (1.1) reduces to(1.5)−Δv+P(x)v=μvw,x∈RN,−Δw+Q(x)w=μ2v2+γw2,x∈RN,v>0,w>0,v,w∈H1RN.$$\begin{equation}\left\{\begin{array}{ll}-\Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right).\end{array}\right. \end{equation}$$Applying the finite dimensional reduction method, Wang and Zhou [22] constructed the infinitely many non-radial positive solutions of (1.5) if the potential functions P(x), Q(x) are radial and satisfy some algebraic decay at infinity. Also, if ε is small, for any positive integer k ≤ N + 1, Tang and Xie [20] proved that (1.1) has a k spikes solution concentrating at some strict local maximum of P(x) and Q(x) by using the finite dimensional reduction provided that |P(x)−P(y)|≤L1|x−y|θ1$|P(x)-P(y)| \leq L_{1}|x-y|^{\theta_{1}}$and |Q(x) − Q(y)| ≤ L2|x − y|θ2 for some positive constants L1, L2, θ1, θ2.Here we want to mention that, very recently, Luo, Peng and Yan [13] revisited the following Schrödinger equation(1.6)−ε2Δu+V(x)u=up−1,u∈H1RN$$\begin{equation}-\varepsilon^{2} \Delta u+V(x) u=u^{p-1}, u \in H^{1}\left(\mathbb{R}^{N}\right) \end{equation}$$with 2 < p < 2*. Under the condition that V(x) obtains its local minimum or local maximum x0 at a closed N − 1 dimensional hypersurface, they obtained the existence of a positive single peak solution for (1.6) concentrating at x0 if x0 is non-degenerate critical point of ΔV and also verified the local uniqueness of single peak solutions by using local Pohazaev type identity.Motivated by [13, 20, 22], we want to apply the finite-dimensional reduction to study the existence of positive single peak solutions for (1.1). Our purpose here is to prove that (1.1) has a single peak solution concentrating at some non-degenerate critical point of Δ(P + Q) on a closed N − 1 dimensional hypersurface.To state our results, throughout this paper, we assume that P(x), Q(x) obtain their local minimum or local maximum at a closed N − 1 dimensional hypersurface Γ. Without loss of generality, we suppose that P(x) = Q(x) = 1 if x ∈ Γ and more precisely, we assume that P(x), Q(x) satisfies the following conditions.(H1)There exist δ > 0 and a closed smooth hypersurface Γ such that if y ∈ Γ, P(y), Q(y) = 1 and P(y), Q(y) > 1 (or P(y), Q(y) < 1) for any y ∈ Wδ \ Γ, where Wδ := {x ∈ ℝN , dist(x, Γ) < δ}.(H2)The level set Γt = {x : P(x), Q(x) = t} is a closed smooth hypersurface for t ∈ [1, 1 +ϑ] (or t ∈ [1−ϑ, 1]) for some small ϑ > 0. Also, for any xt ∈ Γt and x0 ∈ Γ, there holds |νt −ν| ≤ C|xt −x0| and |ςi,t −ςi| ≤ C|xt −x0|, i = 1,2, ··· , N − 1. Here, throughout this paper, we denote by νt(ν) the outward unit normal vector of Γt(Γ) at xt(x0), while we use ςi,t(ςi) to denote the i-th principal tangential unit vector of Γt(Γ) at xt(x0).(H3)For any x ∈ Br(x0), it holds P(x) = Q(x), where x0 ∈ Γ and r is a small positive constant.Remark 1.1Let F(x)=∑i=1Nxi2ai2−1$F(x)=\sum_{i=1}^{N} \frac{x_{i}^{2}}{a_{i}^{2}}-1$with ai>0,ai≠aj(i≠j)$a_{i}>0, a_{i} \neq a_{j}(i \neq j)$and Γ=x∈RN:F(x)=0.$\Gamma=\left\{x \in \mathbb{R}^{N}: F(x)=0\right\}.$TakeP(x)=Q(x)=F2+1, in W0,$$\begin{equation}P(x)=Q(x)=F^{2}+1, \text { in } W_{0}, \end{equation}$$where W0=x∈RN:∑i=1Nxi2ai2−1≤δ0$W_{0}=\left\{x \in \mathbb{R}^{N}:\left|\sum_{i=1}^{N} \frac{x_{i}^{2}}{a_{i}^{2}}-1\right| \leq \delta_{0}\right\}$for some small fixed δ0 > 0. Then we can use the above conditions to the potentials P(x), Q(x).Let us point out that if Γ is a local minimum (or local maximum) set of P(x) and Q(x), then for any x ∈ Γ,P(x)=1,∇P(x)=0$$\begin{equation}P(x)=1, \nabla P(x)=0 \end{equation}$$andQ(x)=1,∇Q(x)=0.$$\begin{equation}Q(x)=1, \nabla Q(x)=0. \end{equation}$$This implies that for any tangential vector ς of Γ at x, one hasDζ∇P(x)=0,Dς∇Q(x)=0,∀x∈Γ,$$\begin{equation}\left(D_{\zeta} \nabla\right) P(x)=0,\left(D_{\varsigma} \nabla\right) Q(x)=0, \forall x \in \Gamma, \end{equation}$$where Dς denotes the directional derivative at the direction ς.Let U be the unique positive radial solution of the following problem(1.7)−Δu+u=u2,u>0, in RN,u(0)=maxx∈RNu(x),u(x)∈H1RN.$$\begin{equation}\left\{\begin{array}{ll}-\Delta u+u=u^{2}, & u>0, \text { in } \mathbb{R}^{N}, \\ u(0)=\max _{x \in \mathbb{R}^{N}} u(x), & u(x) \in H^{1}\left(\mathbb{R}^{N}\right) .\end{array}\right. \end{equation}$$It is well-known in [10] that U(x) is strictly decreasing and its s order derivative satisfiesDsU(x)e|x||x|1−N2≤C$$\begin{equation}\left|D^{s} U(x)\right| e^{|x|}|x|^{\frac{1-N}{2}} \leq C \end{equation}$$for |s| ≤ 1 and some constant C > 0.For xε close to x0, if we denoteVˉε,xε=PxεUPxεx−xεε,$$\begin{equation}\bar{V}_{\varepsilon, x_{\varepsilon}}=P\left(x_{\varepsilon}\right) U\left(\frac{\sqrt{P\left(x_{\varepsilon}\right)}\left(x-x_{\varepsilon}\right)}{\varepsilon}\right), \end{equation}$$then Vˉε,xE$\bar{V}_{\varepsilon, x_{\mathcal{E}}}$solves(1.8)−ε2Δv+Pxεv=v2,v>0,x∈RN,vxε=maxx∈RNv(x),v∈H1RN.$$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P\left(x_{\varepsilon}\right) v=v^{2}, & v>0, \quad x \in \mathbb{R}^{N}, \\ v\left(x_{\varepsilon}\right)=\max _{x \in \mathbb{R}^{N}} v(x), & v \in H^{1}\left(\mathbb{R}^{N}\right) .\end{array}\right. \end{equation}$$Also, writeWˉε,xε=QxεUQxEx−xEε,$$\begin{equation}\bar{W}_{\varepsilon, x_{\varepsilon}}=Q\left(x_{\varepsilon}\right) U\left(\frac{\sqrt{Q\left(x_{\mathcal{E}}\right)}\left(x-x_{\mathcal{E}}\right)}{\varepsilon}\right), \end{equation}$$which is the solution of(1.9)−ε2Δw+Qxεw=w2,w>0,x∈RN,wxE=maxx∈RNw(x),w∈H1RN.$$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta w+Q\left(x_{\varepsilon}\right) w=w^{2}, & w>0, \quad x \in \mathbb{R}^{N}, \\ w\left(x_{\mathcal{E}}\right)=\max _{x \in \mathbb{R}^{N}} w(x), & w \in H^{1}\left(\mathbb{R}^{N}\right) .\end{array}\right. \end{equation}$$Take Vε,xε,Wε,χε=αVˉε,χε,βWˉε,xε with α=1μ2(μ−γ)μ,β=1μ. Then Vε,Xε,Wε,xε$\left(V_{\varepsilon, x_{\varepsilon}}, W_{\varepsilon, \chi_{\varepsilon}}\right)=\left(\alpha \bar{V}_{\varepsilon, \chi_{\varepsilon}}, \beta \bar{W}_{\varepsilon, x_{\varepsilon}}\right) \text { with } \alpha=\frac{1}{\mu} \sqrt{\frac{2(\mu-\gamma)}{\mu}}, \beta=\frac{1}{\mu} . \text { Then }\left(V_{\varepsilon, X_{\varepsilon}}, W_{\varepsilon, x_{\varepsilon}}\right)$solves(1.10)−ε2Δv+Pxεv=μvw,x∈RN,−ε2Δw+Qxεw=μ2v2+γw2,x∈RN,v>0,w>0,v,w∈H1RN$$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P\left(x_{\varepsilon}\right) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q\left(x_{\varepsilon}\right) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right)\end{array}\right. \end{equation}$$since (H3) holds.For ε > 0 small, we will use (Vε,xε ,Wε,xε ) to construct the single peak solutions concentrating at x0. First we give the following definitions.Definition 1.2We say that (vε , wε) is a single peak solution of (1.1) concentrating at x0 if there exist xε∈RN$x_{\mathcal{\varepsilon}} \in \mathbb{R}^{N}$with |xε − x0| = o(1) such thatvε−Vε,Xε,wε−Wε,Xεε=oεN2,$$\begin{equation}\left\|\left(v_{\varepsilon}-V_{\varepsilon, X_{\varepsilon}}, w_{\varepsilon}-W_{\varepsilon, X_{\varepsilon}}\right)\right\|_{\varepsilon}=o\left(\varepsilon^{\frac{N}{2}}\right), \end{equation}$$where ∥(u,v)∥ε=∫RNε2|∇u|2+P(x)u2+ε2|∇v|2+Q(x)v2dx$\|(u, v)\|_{\varepsilon}=\int\limits_{\mathbb{R}^{N}}\left(\varepsilon^{2}|\nabla u|^{2}+P(x) u^{2}+\varepsilon^{2}|\nabla v|^{2}+Q(x) v^{2}\right) d x$Definition 1.3We say that a critical point x0 ∈ Γ of Δ(P + Q) on Γ is non-degenerate if it holds∂2Px0+Qx0∂v2≠0,det∂2ΔPx0∂ςi∂ςj1≤i,j≤N−1+∂2ΔQx0∂ςi∂ςj1≤i,j≤N−1≠0.$$\begin{equation}\frac{\partial^{2}\left(P\left(x_{0}\right)+Q\left(x_{0}\right)\right)}{\partial v^{2}} \neq 0, \quad \operatorname{det}\left(\left(\frac{\partial^{2} \Delta P\left(x_{0}\right)}{\partial \varsigma_{i} \partial \varsigma_{j}}\right)_{1 \leq i, j \leq N-1}+\left(\frac{\partial^{2} \Delta Q\left(x_{0}\right)}{\partial \varsigma_{i} \partial \varsigma_{j}}\right)_{1 \leq i, j \leq N-1}\right) \neq 0. \end{equation}$$The main result of this paper is the following.Theorem 1.4Assume that (H1)− (H3) hold. If x0 ∈ Γ is a non-degenerate critical point of Δ(P + Q), then there exists ε0 > 0 such that (1.1) has a single peak solution (vε , wε) concentrating at x0 provided ε ∈ (0, ε0].As in [13, 20, 22], we mainly apply the finite dimensional reduction method to prove our main result. Compared with [13],we have to overcome much difficulties in the reduction process which involves some technical and careful computations due to the χ(2) nonlinearity appears. Moreover, to our best knowledge, our result exhibits a new phenomenon for the coupled Schrödinger system with χ(2) nonlinearity.Remark 1.5Combining the ideas from [9, 13], where in [9] the coupled nonlinear Gross-Pitaevskii system was studied, we guess that the following conclusions may hold.(1) On the basis of Theorem 1.4, further we can prove the local uniqueness of single peak solutions by using local Pohazaev type identity.(2) Under the conditions of Theorem 1.4, if Δ(P + Q) has an isolated maximum or minimum point x0 ∈ Γ, then for any integer k > 0, (1.1) has a k-peaks solution whose peaks cluster at x0.The structure of this paper is organized as follows. In section 2, we do some preliminaries and then we carry out the finite dimensional reduction. We will prove our main result in section 3. In the sequel, for simplicity of notations we write ∫$\int$f to mean the Lebesgue integral of f (x) in ℝN.2the finite dimensional reductionIn this section, we mainly give some preliminaries and do the finite dimensional reduction. Hereafter, for any function K(x) > 0, we define the Sobolev spaceHε,K1=u∈H1RN:∫ε2|∇u|2+K(x)u2<∞,$$\begin{equation}H_{\varepsilon, K}^{1}=\left\{u \in H^{1}\left(\mathbb{R}^{N}\right): \int\left(\varepsilon^{2}|\nabla u|^{2}+K(x) u^{2}\right)<\infty\right\}, \end{equation}$$endowed with the standard norm∥u∥ε,K=∫ε2|∇u|2+K(x)u212,$$\begin{equation}\|u\|_{\varepsilon, K}=\left(\int\left(\varepsilon^{2}|\nabla u|^{2}+K(x) u^{2}\right)\right)^{\frac{1}{2}}, \end{equation}$$which is induced by the inner product〈u,v〉ε,K=∫ε2∇u∇v+K(x)uv.$$\begin{equation}\langle u, v\rangle_{\varepsilon, K}=\int\left(\varepsilon^{2} \nabla u \nabla v+K(x) u v\right) . \end{equation}$$Now we define H to be the product space Hε,P1×Hε,Q1$H_{\varepsilon, P}^{1} \times H_{\varepsilon, Q}^{1}$with the norm∥(u,v)∥ε2=∥u∥ε,P2+∥v∥ε,Q2$$\begin{equation}\|(u, v)\|_{\varepsilon}^{2}=\|u\|_{\varepsilon, P}^{2}+\|v\|_{\varepsilon, Q}^{2}\end{equation}$$and setEε,xε=(φ,ψ)∈H:(φ,ψ),∂Vε,xε∂xi,∂Wε,Xε∂xiε=0$$\begin{equation}E_{\varepsilon, x_{\varepsilon}}=\left\{(\varphi, \psi) \in H:\left\langle(\varphi, \psi),\left(\frac{\partial V_{\varepsilon, x_{\varepsilon}}}{\partial x_{i}}, \frac{\partial W_{\varepsilon, X_{\varepsilon}}}{\partial x_{i}}\right)\right\rangle_{\varepsilon}=0\right\} \end{equation}$$for i = 1, · · · , N, where(φ,ψ),∂Vε,Xε∂xi,∂Wε,Xε∂xiε=∫ε2∇φ∇∂Vε,xε∂xi+P(x)∂Vε,xε∂xiφ+ε2∇ψ∇∂Wε,xε∂xi+Q(x)∂Wε,Xε∂xiψ.$$\begin{equation}\begin{array}{l}\left\langle(\varphi, \psi),\left(\frac{\partial V_{\varepsilon, X_{\varepsilon}}}{\partial x_{i}}, \frac{\partial W_{\varepsilon, X_{\varepsilon}}}{\partial x_{i}}\right)\right\rangle_{\varepsilon} \\ =\int\left(\varepsilon^{2} \nabla \varphi \nabla \frac{\partial V_{\varepsilon, x_{\varepsilon}}}{\partial x_{i}}+P(x) \frac{\partial V_{\varepsilon, x_{\varepsilon}}}{\partial x_{i}} \varphi+\varepsilon^{2} \nabla \psi \nabla \frac{\partial W_{\varepsilon, x_{\varepsilon}}}{\partial x_{i}}+Q(x) \frac{\partial W_{\varepsilon, X_{\varepsilon}}}{\partial x_{i}} \psi\right) .\end{array} \end{equation}$$Note that the variational functional corresponding to (1.1) is(2.1)Iε(v,w)=12∫ε2|∇v|2+P(x)v2+ε2|∇w|2+Q(x)w2−μ2∫v2w−γ3∫w3.$$\begin{equation}I_{\varepsilon}(v, w)=\frac{1}{2} \int\left(\varepsilon^{2}|\nabla v|^{2}+P(x) v^{2}+\varepsilon^{2}|\nabla w|^{2}+Q(x) w^{2}\right)-\frac{\mu}{2} \int v^{2} w-\frac{\gamma}{3} \int w^{3} . \end{equation}$$Then I ∈ C2(H, ℝ) and its critical points are solutions of (1.1).SetJε(φ,ψ)=IεVε,xε+φ,Wε,xε+ψ,(φ,ψ)∈Eε,xε.$$\begin{equation}J_{\varepsilon}(\varphi, \psi)=I_{\varepsilon}\left(V_{\varepsilon, x_{\varepsilon}}+\varphi, W_{\varepsilon, x_{\varepsilon}}+\psi\right), \quad(\varphi, \psi) \in E_{\varepsilon, x_{\varepsilon}}. \end{equation}$$We can expand Jε(φ, ψ) as follows:(2.2)Jε(φ,ψ)=Jε(0,0)+ℓε(φ,ψ)+12Lε(φ,ψ)+Rε(φ,ψ),$$\begin{equation}J_{\varepsilon}(\varphi, \psi)=J_{\varepsilon}(0,0)+\ell_{\varepsilon}(\varphi, \psi)+\frac{1}{2} L_{\varepsilon}(\varphi, \psi)+R_{\varepsilon}(\varphi, \psi), \end{equation}$$whereℓε(φ,ψ)=∫ε2∇Vε,xε∇φ+P(x)Vε,xεφ+ε2∇Wε,xε∇ψ+Q(x)Wε,xεψ−μ∫Vε,xεWε,xεφ−μ2∫Vε,xε2ψ−γ∫Wε,xε2ψ=∫P(x)−PxεVε,Xεφ+Q(x)−QxεWε,Xεψ,Lε(φ,ψ)=∫ε2|∇φ|2+P(x)φ2+ε2|∇ψ|2+Q(x)ψ2−∫μWε,xεφ2+2γWε,xεψ2+2μVε,xεφψ$$\begin{equation}\begin{array}{c}\ell_{\varepsilon}(\varphi, \psi)=\int\left(\varepsilon^{2} \nabla V_{\varepsilon, x_{\varepsilon}} \nabla \varphi+P(x) V_{\varepsilon, x_{\varepsilon}} \varphi+\varepsilon^{2} \nabla W_{\varepsilon, x_{\varepsilon}} \nabla \psi+Q(x) W_{\varepsilon, x_{\varepsilon}} \psi\right)-\mu \int V_{\varepsilon, x_{\varepsilon}} W_{\varepsilon, x_{\varepsilon}} \varphi \\ -\frac{\mu}{2} \int V_{\varepsilon, x_{\varepsilon}}^{2} \psi-\gamma \int W_{\varepsilon, x_{\varepsilon}}^{2} \psi \\ =\int\left(P(x)-P\left(x_{\varepsilon}\right)\right) V_{\varepsilon, X_{\varepsilon}} \varphi+\left(Q(x)-Q\left(x_{\varepsilon}\right)\right) W_{\varepsilon, X_{\varepsilon}} \psi, \\ L_{\varepsilon}(\varphi, \psi)=\int\left(\varepsilon^{2}|\nabla \varphi|^{2}+P(x) \varphi^{2}+\varepsilon^{2}|\nabla \psi|^{2}+Q(x) \psi^{2}\right)-\int\left(\mu W_{\varepsilon, x_{\varepsilon}} \varphi^{2}+2 \gamma W_{\varepsilon, x_{\varepsilon}} \psi^{2}\right. \\ \left.\quad+2 \mu V_{\varepsilon, x_{\varepsilon}} \varphi \psi\right)\end{array} \end{equation}$$andRε(φ,ψ)=−μ2∫φ2ψ−γ3∫ψ3.$$\begin{equation}R_{\varepsilon}(\varphi, \psi)=-\frac{\mu}{2} \int \varphi^{2} \psi-\frac{\gamma}{3} \int \psi^{3} . \end{equation}$$It follows from [5] that Vε,Xε+φ,Wε,xε+ψ$\left(V_{\varepsilon, X_{\varepsilon}}+\varphi, W_{\varepsilon, x_{\varepsilon}}+\psi\right)$is a critical point of Iε(v, w) if and only if (φ, ψ) is a critical point of Jε(φ, ψ). In order to find a critical point for Jε(φ, ψ), we need to discuss each terms in the expansion (2.2).Lemma 2.1There exists C > 0 independent of ε such thatRεi(φ,ψ)≤Cε−N2∥(φ,ψ)∥ε3−i,i=0,1,2,$$\begin{equation}\left\|R_{\varepsilon}^{i}(\varphi, \psi)\right\| \leq C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}^{3-i}, i=0,1,2, \end{equation}$$where Rεi(φ,ψ)$R_{\varepsilon}^{i}(\varphi, \psi)$denotes the i-th derivative of Rε(φ, ψ).Proof. Recall thatRε(φ,ψ)=−μ2∫φ2ψ−γ3∫ψ3.$$\begin{equation}R_{\varepsilon}(\varphi, \psi)=-\frac{\mu}{2} \int \varphi^{2} \psi-\frac{\gamma}{3} \int \psi^{3} . \end{equation}$$By the direct computations, we getRε′(φ,ψ),(ξ,η)=−μ2∫φ2η+2φψξ−γ∫ψ2η$$\begin{equation}\left\langle R_{\varepsilon}^{\prime}(\varphi, \psi),(\xi, \eta)\right\rangle=-\frac{\mu}{2} \int\left(\varphi^{2} \eta+2 \varphi \psi \xi\right)-\gamma \int \psi^{2} \eta \end{equation}$$andRε′′(φ,ψ)(ξ,η),(g,h)=−μ∫(φgη+φhξ+ψgξ)−2γ∫ψhη.$$\begin{equation}\left\langle R_{\varepsilon}^{\prime \prime}(\varphi, \psi)(\xi, \eta),(g, h)\right\rangle=-\mu \int(\varphi g \eta+\varphi h \xi+\psi g \xi)-2 \gamma \int \psi h \eta. \end{equation}$$Observe that by letting uε(x) = u(εx), it is easy to check(2.3)∥u∥LpRN≤Cε1p−12N∥u∥ε$$\begin{equation}\|u\|_{L^{p}\left(\mathbb{R}^{N}\right)} \leq C \varepsilon^{\left(\frac{1}{p}-\frac{1}{2}\right) N}\|u\|_{\varepsilon} \end{equation}$$with ∥u∥ε=∫ε2|∇u|2+u212 for 2≤p≤2⋆$\|u\|_{\varepsilon}=\left(\int\left(\varepsilon^{2}|\nabla u|^{2}+u^{2}\right)\right)^{\frac{1}{2}} \text { for } 2 \leq p \leq 2^{\star}$and some C > 0. Then we haveRE(φ,ψ)≤C∫|φ|2|ψ|+∫|ψ|3≤C∫|φ|323∫|ψ|313+C∫|ψ|3≤Cε−N2∥φ∥ε2∥ψ∥ε+Cε−N2∥φ∥ε3≤Cε−N2∥(φ,ψ)∥ε3,$$\begin{equation}\begin{aligned}\left|R_{\mathcal{E}}(\varphi, \psi)\right| & \leq C\left(\int|\varphi|^{2}|\psi|+\int|\psi|^{3}\right) \\ & \leq C\left(\int|\varphi|^{3}\right)^{\frac{2}{3}}\left(\int|\psi|^{3}\right)^{\frac{1}{3}}+C \int|\psi|^{3} \\ & \leq C \varepsilon^{-\frac{N}{2}}\|\varphi\|_{\varepsilon}^{2}\|\psi\|_{\varepsilon}+C \varepsilon^{-\frac{N}{2}}\|\varphi\|_{\varepsilon}^{3} \\ & \leq C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}^{3}, \end{aligned} \end{equation}$$Rε′(φ,ψ),(ξ,η)≤C∫|φ|2|η|+|φ||ψ||ξ|+|ψ|2|η|≤C∫|φ|323∫|η|313+C∫|φ|313∫|ψ|313∫|ξ|313+C∫|ψ|323∫|η|313≤Cε−N2∥φ∥ε2∥η∥ε+∥φ∥ε∥ψ∥ε∥ξ∥ε+∥ψ∥ε2∥η∥ε≤Cε−N2∥(φ,ψ)∥ε2∥(ξ,η)∥ε$$\begin{equation}\begin{array}{l}\left|R_{\varepsilon}^{\prime}(\varphi, \psi),(\xi, \eta)\right| \\ \leq C \int\left(|\varphi|^{2}|\eta|+|\varphi||\psi||\xi|+|\psi|^{2}|\eta|\right) \\ \leq C\left(\int|\varphi|^{3}\right)^{\frac{2}{3}}\left(\int|\eta|^{3}\right)^{\frac{1}{3}}+C\left(\int|\varphi|^{3}\right)^{\frac{1}{3}}\left(\int|\psi|^{3}\right)^{\frac{1}{3}}\left(\int|\xi|^{3}\right)^{\frac{1}{3}}+C\left(\int|\psi|^{3}\right)^{\frac{2}{3}}\left(\int|\eta|^{3}\right)^{\frac{1}{3}} \\ \leq C \varepsilon^{-\frac{N}{2}}\left(\|\varphi\|_{\varepsilon}^{2}\|\eta\|_{\varepsilon}+\|\varphi\|_{\varepsilon}\|\psi\|_{\varepsilon}\|\xi\|_{\varepsilon}+\|\psi\|_{\varepsilon}^{2}\|\eta\|_{\varepsilon}\right) \\ \leq C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}^{2}\|(\xi, \eta)\|_{\varepsilon}\end{array} \end{equation}$$and similarly,Rε′′(φ,ψ)(ξ,η),(g,h)≤C∫(|φ||g||η|+|φ||h||ξ|+|ψ||g||ξ|+|ψ||h||η|)≤Cε−N2∥(φ,ψ)∥ε∥(ξ,η)∥ε∥(g,h)∥ε.$$\begin{equation}\begin{array}{l}\left|R_{\varepsilon}^{\prime \prime}(\varphi, \psi)(\xi, \eta),(g, h)\right| \\ \leq C \int(|\varphi||g||\eta|+|\varphi||h||\xi|+|\psi||g||\xi|+|\psi||h||\eta|) \\ \leq C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}\|(\xi, \eta)\|_{\varepsilon}\|(g, h)\|_{\varepsilon}.\end{array} \end{equation}$$This completes our proof.Lemma 2.2There exists C > 0 independent of ε such thatℓε≤C∇Pxε+∇QxεεN2+1+εN2+2.$$\begin{equation}\left\|\ell_{\varepsilon}\right\| \leq C\left(\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right). \end{equation}$$Proof. For any (φ,ψ)∈Eε,xε,$(\varphi, \psi) \in E_{\varepsilon, x_{\varepsilon}},$taking into account the decay property of U, we find for any fixed d > 0,ℓε(φ,ψ)=∫P(x)−PxεVε,xεφ+Q(x)−QxεWε,xεψ=εN∫Pεy+xε−PxεαPxεUPxεyφεy+xε+∫Qεy+xε−QxεβQxεUQxεyψεy+xε=εN∫Bdε(0)Pεy+xε−PxεαPxεUPxεyφεy+xε+∫Bdε(0)Qεy+xε−QxεβQxεUQxεyψεy+χε+∫RN∖Bdε(0)Pεy+xε−PxεαPxεUPxεyφεy+xε$$\begin{equation}\begin{aligned} \ell_{\varepsilon}(\varphi, \psi)=& \int\left(P(x)-P\left(x_{\varepsilon}\right)\right) V_{\varepsilon, x_{\varepsilon}} \varphi+\left(Q(x)-Q\left(x_{\varepsilon}\right)\right) W_{\varepsilon, x_{\varepsilon}} \psi \\=& \varepsilon^{N}\left(\int\left(P\left(\varepsilon y+x_{\varepsilon}\right)-P\left(x_{\varepsilon}\right)\right) \alpha P\left(x_{\varepsilon}\right) U\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right) \varphi\left(\varepsilon y+x_{\varepsilon}\right)\right.\\ &\left.\quad+\int\left(Q\left(\varepsilon y+x_{\varepsilon}\right)-Q\left(x_{\varepsilon}\right)\right) \beta Q\left(x_{\varepsilon}\right) U\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right) \psi\left(\varepsilon y+x_{\varepsilon}\right)\right) \\ =\varepsilon^{N} &\left\{\int_{B_{\frac{d}{\varepsilon}}(0)}\left(P\left(\varepsilon y+x_{\varepsilon}\right)-P\left(x_{\varepsilon}\right)\right) \alpha P\left(x_{\varepsilon}\right) U\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right) \varphi\left(\varepsilon y+x_{\varepsilon}\right)\right.\\ &+\int_{B_{\frac{d}{\varepsilon}}(0)}\left(Q\left(\varepsilon y+x_{\varepsilon}\right)-Q\left(x_{\varepsilon}\right)\right) \beta Q\left(x_{\varepsilon}\right) U\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right) \psi\left(\varepsilon y+\chi_{\varepsilon}\right) \\ &+\int_{\mathbb{R}^{N} \backslash B_{\frac{d}{\varepsilon}}(0)}\left(P\left(\varepsilon y+x_{\varepsilon}\right)-P\left(x_{\varepsilon}\right)\right) \alpha P\left(x_{\varepsilon}\right) U\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right) \varphi\left(\varepsilon y+x_{\varepsilon}\right) \end{aligned} \end{equation}$$+∫RN∖Bdε(0)Qεy+xε−QxεβQxεUQxεyψεy+xε≤CεN2∫Bdε(0)α2Pεy+xε−Pxε2P2xεU2Pxεy12+∫Bdε(0)β2Qεy+xε−Qxε2Q2xεU2Qxεy12∥(φ,ψ)∥ε+CεN2∬RN∖Bdε(0)U2Pxεy12+∫RN∖Bdε(0)U2Qxεy12∥(φ,ψ)∥ε=OεN2e−τε∥(φ,ψ)∥ε+OεN2∫dε∇Pxε|εy|+|εy|22U2Pxεy12+εN2∫Bdε(0)∇Qxε|εy|+|εy|22U2Qxεy12∥(φ,ψ)∥ε=O∇Pxε+∇QxεεN2+1+εN2+2∥(φ,ψ)∥ε$$\begin{equation}\begin{array}{c}\left.+\int_{\mathbb{R}^{N} \backslash B_{\frac{d}{\varepsilon}}(0)}\left(Q\left(\varepsilon y+x_{\varepsilon}\right)-Q\left(x_{\varepsilon}\right)\right) \beta Q\left(x_{\varepsilon}\right) U\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right) \psi\left(\varepsilon y+x_{\varepsilon}\right)\right\} \\ \leq C \varepsilon^{\frac{N}{2}}\left[\left(\int_{B_{\frac{d}{\varepsilon}}(0)} \alpha^{2}\left(P\left(\varepsilon y+x_{\varepsilon}\right)-P\left(x_{\varepsilon}\right)\right)^{2} P^{2}\left(x_{\varepsilon}\right) U^{2}\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right. \\ \left.+\left(\int_{B_{\frac{d}{\varepsilon}}(0)} \beta^{2}\left(Q\left(\varepsilon y+x_{\mathcal{\varepsilon}}\right)-Q\left(x_{\varepsilon}\right)\right)^{2} Q^{2}\left(x_{\varepsilon}\right) U^{2}\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right]\|(\varphi, \psi)\|_{\varepsilon} \\ +C \varepsilon^{\frac{N}{2}}\left[\left(\iint_{\mathbb{R}^{N} \backslash B_{\frac{d}{\varepsilon}}(0)} U^{2}\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}+\left(\int_{\mathbb{R}^{N} \backslash B_{\frac{d}{\varepsilon}}(0)} U^{2}\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right]\|(\varphi, \psi)\|_{\varepsilon} \\ =O\left(\varepsilon^{\frac{N}{2}} e^{-\frac{\tau}{\varepsilon}}\right)\|(\varphi, \psi)\|_{\varepsilon}+O\left[\varepsilon^{\frac{N}{2}}\left(\int_{\frac{d}{\varepsilon}}\left(\left|\nabla P\left(x_{\varepsilon}\right)\right||\varepsilon y|+|\varepsilon y|^{2}\right)^{2} U^{2}\left(\sqrt{P\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right. \\ \left.\quad+\varepsilon^{\frac{N}{2}}\left(\int_{B_{\frac{d}{\varepsilon}}(0)}\left(\left|\nabla Q\left(x_{\varepsilon}\right)\right||\varepsilon y|+|\varepsilon y|^{2}\right)^{2} U^{2}\left(\sqrt{Q\left(x_{\varepsilon}\right)} y\right)\right)^{\frac{1}{2}}\right]\|(\varphi, \psi)\|_{\varepsilon} \\ =O\left(\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right)\|(\varphi, \psi)\|_{\varepsilon}\end{array} \end{equation}$$for some τ > 0.Now it is easy to check that∫ε2(∇v∇φ+∇w∇ψ)+P(x)vφ+Q(x)wψ−∫μWε,xεvφ+μVε,xεvψ+μVε,xεwφ+2γWε,xεwψ$$\begin{equation}\begin{array}{c}\int\left(\varepsilon^{2}(\nabla v \nabla \varphi+\nabla w \nabla \psi)+P(x) v \varphi+Q(x) w \psi\right)-\int\left(\mu W_{\varepsilon, x_{\varepsilon}} v \varphi+\mu V_{\varepsilon, x_{\varepsilon}} v \psi+\mu V_{\varepsilon, x_{\varepsilon}} w \varphi\right. \\ \left.+2 \gamma W_{\varepsilon, x_{\varepsilon}} w \psi\right)\end{array} \end{equation}$$is a bounded bi-linear functional in Eε,xε.$E_{\varepsilon, x_{\varepsilon}}.$Hence we can define L to be a bounded linear map from Eε,xε$E_{\varepsilon, x_{\varepsilon}}$to Eε,xε such that for any (v,w),(φ,ψ)∈Eε,χε,$(v, w),(\varphi, \psi) \in E_{\varepsilon, \chi_{\varepsilon}},$〈L(v,w),(φ,ψ)〉=∫ε2(∇v∇φ+∇w∇ψ)+P(x)vφ+Q(x)wψ−∫μWε,xεvφ+μVε,xεvψ+μVε,xεwφ+2γWε,xεwψ.$$\begin{equation}\begin{aligned}\langle L(v, w),(\varphi, \psi)\rangle=\int &\left(\varepsilon^{2}(\nabla v \nabla \varphi+\nabla w \nabla \psi)+P(x) v \varphi+Q(x) w \psi\right) \\ &-\int\left(\mu W_{\varepsilon, x_{\varepsilon}} v \varphi+\mu V_{\varepsilon, x_{\varepsilon}} v \psi+\mu V_{\varepsilon, x_{\varepsilon}} w \varphi+2 \gamma W_{\varepsilon, x_{\varepsilon}} w \psi\right). \end{aligned} \end{equation}$$Next we want to discuss the invertibility of L in Eε,Xε.$E_{\varepsilon, X_{\varepsilon}}.$To this end, we denote (V,W) = (αU, βU)). Then (V,W) solves(2.4)−Δv+v=μvw,x∈RN,−Δw+w=μ2v2+γw2,x∈RN.$$\begin{equation}\left\{\begin{array}{ll}-\Delta v+v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\Delta w+w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}.\end{array}\right. \end{equation}$$It follows from Proposition 2.2 in [22] thatProposition 2.3For any μ > 0 and μ > γ, (V,W) is non-degenerate for the system (2.4) in H1RN×H1RN$H^{1}\left(\mathbb{R}^{N}\right) \times H^{1}\left(\mathbb{R}^{N}\right)$in the sence that the kernel is given by span λ(μ,γ)∂V∂xi,∂W∂xi∣i=1,2,⋯,N$\left\{\lambda(\mu, \gamma) \frac{\partial V}{\partial x_{i}}, \frac{\partial W}{\partial x_{i}} \mid i=1,2, \cdots, N\right\}$, where λ(μ,γ)≠0.$\lambda(\mu, \gamma) \neq 0.$Using the above result, we come to discuss the invertibility of L in Eε,Xε.$E_{\varepsilon, X_{\varepsilon}}.$Lemma 2.4There exist constants ρ > 0 and ε0 > 0 such that for all ε ∈ (0, ε0],∥L(v,w)∥≥ρ∥(v,w)∥ε,∀(v,w)∈Eε,xε.$$\begin{equation}\|L(v, w)\| \geq \rho\|(v, w)\|_{\varepsilon}, \quad \forall(v, w) \in E_{\varepsilon, x_{\varepsilon}}. \end{equation}$$Proof. We argue by contradiction. Suppose that there exist εn → 0, χεn$\chi_{\varepsilon_{n}}$→ x0 and (vn , wn) ∈ Eεn,xεn$E_{\varepsilon_{n}, x_{\varepsilon_{n}}}$such that for any φn,ψn∈Eεn,xεn,$\left(\varphi_{n}, \psi_{n}\right) \in E_{\varepsilon_{n}, x_{\varepsilon_{n}}},$(2.5)Lvn,wn,φn,ψn=on(1)vn,wnεnφn,ψnεn.$$\begin{equation}\left\langle L\left(v_{n}, w_{n}\right),\left(\varphi_{n}, \psi_{n}\right)\right\rangle=o_{n}(1)\left\|\left(v_{n}, w_{n}\right)\right\|_{\varepsilon_{n}}\left\|\left(\varphi_{n}, \psi_{n}\right)\right\|_{\varepsilon_{n}}. \end{equation}$$Without loss of generality, we can assume that vn,wnεn2=εnN$\left\|\left(v_{n}, w_{n}\right)\right\|_{\varepsilon_{n}}^{2}=\varepsilon_{n}^{N}$and letv˜n(x)=1PxεnvnεnχPxεn+χεn$$\begin{equation}\widetilde{v}_{n}(x)=\frac{1}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}} v_{n}\left(\frac{\varepsilon_{n} \chi}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+\chi_{\varepsilon_{n}}\right) \end{equation}$$andw˜n(x)=1QxεnwnεnxQxεn+χεn.$$\begin{equation}\widetilde{w}_{n}(x)=\frac{1}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}} w_{n}\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+\chi_{\varepsilon_{n}}\right). \end{equation}$$Then, in view of vn,wnεn2=εnN,$\left\|\left(v_{n}, w_{n}\right)\right\|_{\varepsilon_{n}}^{2}=\varepsilon_{n}^{N},$we getv˜n,w˜nH1×H1=∫∇v˜n2+v˜n2+∇w˜n2+w˜n2≤C,$$\begin{equation}\left\|\left(\widetilde{v}_{n}, \widetilde{w}_{n}\right)\right\|_{H^{1} \times H^{1}}=\int\left(\left|\nabla \widetilde{v}_{n}\right|^{2}+\widetilde{v}_{n}^{2}+\left|\nabla \widetilde{w}_{n}\right|^{2}+\widetilde{w}_{n}^{2}\right) \leq C, \end{equation}$$which implies that up to a subsequence, there exists v, w ∈ H1(ℝN) such that as n → +∞,v˜n⇀v,w˜n⇀w in H1RN,v˜n→v,w˜n→w in Lloc2RN.$$\begin{equation}\left\{\begin{array}{lll}\widetilde{v}_{n} \rightharpoonup v, & \widetilde{w}_{n} \rightharpoonup w & \text { in } H^{1}\left(\mathbb{R}^{N}\right), \\ \widetilde{v}_{n} \rightarrow v, & \widetilde{w}_{n} \rightarrow w & \text { in } L_{l o c}^{2}\left(\mathbb{R}^{N}\right).\end{array}\right. \end{equation}$$Now we claim that v = w = 0. Considering (2.5), we find(2.6)∫P2xεn∇v˜n∇φ˜n+∇w˜n∇ψ˜n−μ(βU)v˜nφ˜n−μ(αU)w˜nφ˜n−μ(αU)v˜nψ˜n−2γ(βU)w˜nψ˜n+PxεnPεnyPxεn+xεnv˜nφ˜n+QxεnQεnyQxεn+xεnw˜nψ˜n=PxεnN4on(1)∫P2xεn∇φ˜n2+Q2xεn∇ψ˜n2+PxεnPεnyPxεn+xεnφ˜n2+QxεnQεnyPxεn+xεnψ˜n212,$$\begin{equation}\begin{aligned} \int\left[P^{2}\left(x_{\varepsilon_{n}}\right)\right.&\left(\nabla \widetilde{v}_{n} \nabla \widetilde{\varphi}_{n}+\nabla \widetilde{w}_{n} \nabla \widetilde{\psi}_{n}-\mu(\beta U) \widetilde{v}_{n} \widetilde{\varphi}_{n}-\mu(\alpha U) \widetilde{w}_{n} \widetilde{\varphi}_{n}-\mu(\alpha U) \widetilde{v}_{n} \widetilde{\psi}_{n}\right.\\ &\left.-2 \gamma(\beta U) \widetilde{w}_{n} \widetilde{\psi}_{n}\right)+P\left(x_{\varepsilon_{n}}\right) P\left(\frac{\varepsilon_{n} y}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{v}_{n} \widetilde{\varphi}_{n} \\ &\left.+Q\left(x_{\varepsilon_{n}}\right) Q\left(\frac{\varepsilon_{n} y}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{w}_{n} \widetilde{\psi}_{n}\right] \\=\left(P\left(x_{\varepsilon_{n}}\right)\right)^{\frac{N}{4}} & o_{n}(1)\left[\int P^{2}\left(x_{\varepsilon_{n}}\right)\left|\nabla \widetilde{\varphi}_{n}\right|^{2}+Q^{2}\left(x_{\varepsilon_{n}}\right)\left|\nabla \widetilde{\psi}_{n}\right|^{2}+P\left(x_{\varepsilon_{n}}\right) P\left(\frac{\varepsilon_{n} y}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{\varphi}_{n}^{2}\right.\\ &\left.+Q\left(x_{\varepsilon_{n}}\right) Q\left(\frac{\varepsilon_{n} y}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{\psi}_{n}^{2}\right]^{\frac{1}{2}}, \end{aligned} \end{equation}$$whereφ˜n(x)=1PxεnφnεnXPxεn+xEn,ψ˜n(x)=1QxεnψnεnxQxεn+χεn$$\begin{equation}\begin{array}{l}\widetilde{\varphi}_{n}(x)=\frac{1}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}} \varphi_{n}\left(\frac{\varepsilon_{n} X}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right), \\ \widetilde{\psi}_{n}(x)=\frac{1}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}} \psi_{n}\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+\chi_{\varepsilon_{n}}\right)\end{array} \end{equation}$$and φ˜n,ψ˜n∈E˜εn,xεn$\left(\widetilde{\varphi}_{n}, \widetilde{\psi}_{n}\right) \in \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}$withE˜εn,xεn=(φ˜,ψ˜):φ˜Pxεnx−xEnεn,ψ˜Qxεnx−xεnεn∈Eεn,xεn.$$\begin{equation}\widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}=\left\{(\widetilde{\varphi}, \widetilde{\psi}):\left(\widetilde{\varphi}\left(\frac{\sqrt{P\left(x_{\varepsilon_{n}}\right)}\left(x-x_{\mathcal{E}_{n}}\right)}{\varepsilon_{n}}\right), \widetilde{\psi}\left(\frac{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}\left(x-x_{\varepsilon_{n}}\right)}{\varepsilon_{n}}\right)\right) \in E_{\varepsilon_{n}, x_{\varepsilon_{n}}}\right\}. \end{equation}$$On the other hand, being vn,wnεn2=εnN,$\left\|\left(v_{n}, w_{n}\right)\right\|_{\varepsilon_{n}}^{2}=\varepsilon_{n}^{N},$we have(2.7)∫Pxεn∇v˜n2+∇w˜n2+PεnχPxεn+xεnv˜n2+QεnxQxεn+xεnw˜n2=PxεnN2−1.$$\begin{equation}\begin{array}{c}\int\left[P\left(x_{\varepsilon_{n}}\right)\left(\left|\nabla \widetilde{v}_{n}\right|^{2}+\left|\nabla \widetilde{w}_{n}\right|^{2}\right)+P\left(\frac{\varepsilon_{n} \chi}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{v}_{n}^{2}\right. \\ \left.+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{w}_{n}^{2}\right]=\left(P\left(x_{\varepsilon_{n}}\right)\right)^{\frac{N}{2}-1}.\end{array} \end{equation}$$So, taking φn=vn,ψn=wn,$\varphi_{n}=v_{n}, \psi_{n}=w_{n},$(2.6) gives(2.8)∫Pxεn∇v˜n2+∇w˜n2−μ(βU)v˜n2−2μ(αU)v˜nw˜n−2γ(βU)w˜n2+PεnxPxεn+xEnv˜n2+QεnxQxεn+xEnw˜n2=on(1)PxεnN2−1.$$\begin{equation}\begin{array}{c}\int\left[P\left(x_{\varepsilon_{n}}\right)\left(\left|\nabla \widetilde{v}_{n}\right|^{2}+\left|\nabla \widetilde{w}_{n}\right|^{2}-\mu(\beta U) \widetilde{v}_{n}^{2}-2 \mu(\alpha U) \widetilde{v}_{n} \widetilde{w}_{n}-2 \gamma(\beta U)\left|\widetilde{w}_{n}\right|^{2}\right)\right. \\ \left.+P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right) \widetilde{v}_{n}^{2}+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right) \widetilde{w}_{n}^{2}\right] \\ =o_{n}(1)\left(P\left(x_{\varepsilon_{n}}\right)\right)^{\frac{N}{2}-1}.\end{array} \end{equation}$$Also, since v˜n,w˜n∈E˜εn,xεn$\left(\widetilde{v}_{n}, \widetilde{w}_{n}\right) \in \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}$, we obtain∫Pxεn∇v˜n∇∂(αU)∂yi+PεnxPxεn+xεnv˜n∂(αU)∂yi+Qxεn∇w˜n∇∂(βU)∂yi+QεnxQxεn+xεnw˜n∂(βU)∂yi=0,$$\begin{equation}\begin{array}{l}\int\left[P\left(x_{\varepsilon_{n}}\right) \nabla \widetilde{v}_{n} \nabla \frac{\partial(\alpha U)}{\partial y_{i}}+P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{v}_{n} \frac{\partial(\alpha U)}{\partial y_{i}}\right. \\ \left.\quad+Q\left(x_{\varepsilon_{n}}\right) \nabla \widetilde{w}_{n} \nabla \frac{\partial(\beta U)}{\partial y_{i}}+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{w}_{n} \frac{\partial(\beta U)}{\partial y_{i}}\right] \\ =0,\end{array} \end{equation}$$which gives that by letting n → +∞,(2.9)∫∇v∇∂(αU)∂xi+v(x)∂(αU)∂xi+∇w∇∂(βU)∂xi+w(x)∂(βU)∂xi=0.$$\begin{equation}\int\left[\nabla v \nabla \frac{\partial(\alpha U)}{\partial x_{i}}+v(x) \frac{\partial(\alpha U)}{\partial x_{i}}+\nabla w \nabla \frac{\partial(\beta U)}{\partial x_{i}}+w(x) \frac{\partial(\beta U)}{\partial x_{i}}\right]=0 . \end{equation}$$Now we take (φ˜,ψ˜)∈C0∞RN×C0∞RN$(\widetilde{\varphi}, \widetilde{\psi}) \in C_{0}^{\infty}\left(\mathbb{R}^{N}\right) \times C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$satisfying(2.10)∫∇φ˜∇∂(αU)∂xi+φ˜∂(αU)∂xi+∇ψ˜∇∂(βU)∂xi+ψ˜∂(βU)∂xi=0.$$\begin{equation}\int\left[\nabla \widetilde{\varphi} \nabla \frac{\partial(\alpha U)}{\partial x_{i}}+\widetilde{\varphi} \frac{\partial(\alpha U)}{\partial x_{i}}+\nabla \widetilde{\psi} \nabla \frac{\partial(\beta U)}{\partial x_{i}}+\widetilde{\psi} \frac{\partial(\beta U)}{\partial x_{i}}\right]=0. \end{equation}$$Meanwhile, we can decompose (φ˜,ψ˜)∈E˜εn,xεn$(\widetilde{\varphi}, \widetilde{\psi}) \in \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}$as follows(φ˜,ψ˜)=φ˜n,ψ˜n−∑i=1Nai,n∂Vεn,λεnεnXPxεn+xεn∂xi,∂Wεn,xεnεnXQxεn+xεn∂xi.$$\begin{equation}(\widetilde{\varphi}, \widetilde{\psi})=\left(\widetilde{\varphi}_{n}, \widetilde{\psi}_{n}\right)-\sum_{i=1}^{N} a_{i, n}\left(\frac{\partial V_{\varepsilon_{n}, \lambda_{\varepsilon_{n}}}\left(\frac{\varepsilon_{n} X}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right)}{\partial x_{i}}, \frac{\partial W_{\varepsilon_{n}, x_{\varepsilon_{n}}}\left(\frac{\varepsilon_{n} X}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right)}{\partial x_{i}}\right) . \end{equation}$$Then from φ˜n,ψ˜n∈E˜εn,xεn$\left(\widetilde{\varphi}_{n}, \widetilde{\psi}_{n}\right) \in \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}}$and the definition ofE˜εn,xεn,$\widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}},$we get(2.11)∫Pxεn∇φ˜∇∂(αU)∂xi+PεnxPxεn+xεnφ˜∂(αU)∂xi+Qxεn∇ψ˜∇∂(βU)∂xi+QεnxQxεn+xεnψ˜∂(βU)∂xi+∫∑i=1Nai,n∇∂(αU)∂xi∇∂(αU)∂xi+PεnxPxεn+xεn∑i=1Nai,n∂(αU)∂xi∂(αU)∂xi+∑i=1Nai,n∇∂(βU)∂xi∇∂(βU)∂xi+QεnxQxεn+xεn∑i=1Nai,n∂(βU)∂xi∂(βU)∂xi=0.$$\begin{equation}\begin{array}{c}\int\left[P\left(x_{\varepsilon_{n}}\right) \nabla \widetilde{\varphi} \nabla \frac{\partial(\alpha U)}{\partial x_{i}}+P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{\varphi} \frac{\partial(\alpha U)}{\partial x_{i}}+Q\left(x_{\varepsilon_{n}}\right) \nabla \widetilde{\psi} \nabla \frac{\partial(\beta U)}{\partial x_{i}}\right. \\ \left.\quad+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \widetilde{\psi} \frac{\partial(\beta U)}{\partial x_{i}}\right]+\int\left[\sum_{i=1}^{N} a_{i, n} \nabla \frac{\partial(\alpha U)}{\partial x_{i}} \nabla \frac{\partial(\alpha U)}{\partial x_{i}}\right. \\ +P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \sum_{i=1}^{N} a_{i, n} \frac{\partial(\alpha U)}{\partial x_{i}} \frac{\partial(\alpha U)}{\partial x_{i}}+\sum_{i=1}^{N} a_{i, n} \nabla \frac{\partial(\beta U)}{\partial x_{i}} \nabla \frac{\partial(\beta U)}{\partial x_{i}} \\ \left.+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\varepsilon_{n}}\right) \sum_{i=1}^{N} a_{i, n} \frac{\partial(\beta U)}{\partial x_{i}} \frac{\partial(\beta U)}{\partial x_{i}}\right]=0.\end{array} \end{equation}$$But by decay property of U, there exists C > 0 such that(2.12)∫∇∂(αU)∂xi2+∇∂(βU)∂xi2+∂(αU)∂xi2+∂(βU)∂xi2≥C>0.$$\begin{equation}\int\left[\left|\nabla \frac{\partial(\alpha U)}{\partial x_{i}}\right|^{2}+\left|\nabla \frac{\partial(\beta U)}{\partial x_{i}}\right|^{2}+\left|\frac{\partial(\alpha U)}{\partial x_{i}}\right|^{2}+\left|\frac{\partial(\beta U)}{\partial x_{i}}\right|^{2}\right] \geq C>0 . \end{equation}$$Hence, combining (2.10)-(2.12), we can easily check that ai,n → 0as n → +∞and then it follows from (2.6) that(2.13)∫[P2(x0)(∇v∇φ˜+∇w∇ψ˜+vφ˜+wψ˜−μ(βU)vφ˜−μ(αU)wφ˜−μ(αU)vψ˜−2γ(βU)wψ˜)]=0.$$\begin{aligned}\int \Big[P^2(x_0&)\big(\nabla v\nabla\widetilde{\varphi}+\nabla w\nabla\widetilde{\psi}+v\widetilde{\varphi}+w\widetilde{\psi} -\mu(\beta U)v\widetilde{\varphi}-\mu(\alpha U)w\widetilde{\varphi}\\ &-\mu(\alpha U)v\widetilde{\psi}-2\gamma(\beta U)w\widetilde{\psi}\big)\Big]=0. \end{aligned}$$This implies that for any (φ,ψ)∈C0∞RN×C0∞RN∩E˜εn,xεn,$(\varphi, \psi) \in C_{0}^{\infty}\left(\mathbb{R}^{N}\right) \times C_{0}^{\infty}\left(\mathbb{R}^{N}\right) \cap \widetilde{E}_{\varepsilon_{n}, x_{\varepsilon_{n}}},$(2.14)∫[∇v∇φ+∇w∇ψ+vφ+wψ−μ(βU)vφ−μ(αU)wφ−μ(αU)vψ−2γ(βU)wψ]=0.$$\begin{equation}\begin{array}{l}\int[\nabla v \nabla \varphi+\nabla w \nabla \psi+v \varphi+w \psi-\mu(\beta U) v \varphi-\mu(\alpha U) w \varphi \\ \quad-\mu(\alpha U) v \psi-2 \gamma(\beta U) w \psi]=0.\end{array} \end{equation}$$On the other hand, from the fact that (αU, βU) solves (2.4), we see(2.15)∫∇v∇∂(αU)∂xi+∇w∇∂(βU)∂xi+v∂(αU)∂xi+w∂(βU)∂xi−μ(βU)v∂(αU)∂xi−μ(αU)w∂(αU)∂xi−μ(αU)v∂(βU)∂xi−2γ(βU)w∂(βU)∂xi=0,$$\begin{equation}\begin{aligned} \int\left[\nabla v \nabla \frac{\partial(\alpha U)}{\partial x_{i}}\right.&+\nabla w \nabla \frac{\partial(\beta U)}{\partial x_{i}}+v \frac{\partial(\alpha U)}{\partial x_{i}}+w \frac{\partial(\beta U)}{\partial x_{i}}-\mu(\beta U) v \frac{\partial(\alpha U)}{\partial x_{i}}-\mu(\alpha U) w \frac{\partial(\alpha U)}{\partial x_{i}} \\\left.-\mu(\alpha U) v \frac{\partial(\beta U)}{\partial x_{i}}-2 \gamma(\beta U) w \frac{\partial(\beta U)}{\partial x_{i}}\right]=0, \end{aligned} \end{equation}$$which, together with (2.14), yields that for any (φ,ψ)∈C0∞RN×C0∞RN$(\varphi, \psi) \in C_{0}^{\infty}\left(\mathbb{R}^{N}\right) \times C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$∫[∇v∇φ+∇w∇ψ+vφ+wψ−μ(βU)vφ−μ(αU)wφ−μ(αU)vψ−2γ(βU)wψ]=0.$$\begin{equation}\begin{array}{l}\int[\nabla v \nabla \varphi+\nabla w \nabla \psi+v \varphi+w \psi-\mu(\beta U) v \varphi-\mu(\alpha U) w \varphi \\ \quad-\mu(\alpha U) v \psi-2 \gamma(\beta U) w \psi]=0.\end{array} \end{equation}$$So (v, w) is a solution of−Δv+v=μ(βU)v+μ(αU)w,x∈RN,−Δw+w=μ(αU)v+2γ(βU)w,x∈RN.$$\begin{equation}\left\{\begin{aligned}-\Delta v+v &=\mu(\beta U) v+\mu(\alpha U) w, & & x \in \mathbb{R}^{N}, \\-\Delta w+w &=\mu(\alpha U) v+2 \gamma(\beta U) w, & & x \in \mathbb{R}^{N} . \end{aligned}\right. \end{equation}$$Using Proposition 2.3, there exist bi ∈ ℝ, i = 1, 2, · · · , N such that(v,w)=∑i=1Nbi∂(αU)∂xi,∂(βU)∂xi.$$\begin{equation}(v, w)=\sum_{i=1}^{N} b_{i}\left(\frac{\partial(\alpha U)}{\partial x_{i}}, \frac{\partial(\beta U)}{\partial x_{i}}\right). \end{equation}$$But (2.9) gives that bi = 0, i = 1, 2, · · · , N. That is (v, w) = (0, 0), which is exactly our claim. Finally, taking into account that v˜n→0$\widetilde{v}_{n} \rightarrow 0$in Lloc2RN$L_{l o c}^{2}\left(\mathbb{R}^{N}\right)$and the exponential decay of U, we have∫Uv˜n2=∫BR(0)Uv˜n2+∫RN∣BR(0)Uv˜n2=oR(1)+Oe−R,$$\begin{equation}\int U\left|\widetilde{v}_{n}\right|^{2}=\int_{B_{R}(0)} U\left|\widetilde{v}_{n}\right|^{2}+\int_{\mathbb{R}^{N} \mid B_{R}(0)} U\left|\widetilde{v}_{n}\right|^{2}=o_{R}(1)+O\left(e^{-R}\right), \end{equation}$$where oR(1) → 0 as R → +∞.As a result, from (2.8) and (2.7), we deduce thaton(1)PxεnN2−1=∫Pxεn∇v˜n2+∇w˜n2−μ(βU)v˜n2−2μ(αU)v˜nw˜n−2γ(βU)w˜n2+PεnxPxεn+xEnv˜n2+QεnxQxεn+xEnw˜n2=PxEnN2−1+oR(1)+Oe−R,$$\begin{equation}\begin{array}{c}o_{n}(1)\left(P\left(x_{\varepsilon_{n}}\right)\right)^{\frac{N}{2}-1}=\int\left[P\left(x_{\varepsilon_{n}}\right)\left(\left|\nabla \widetilde{v}_{n}\right|^{2}+\left|\nabla \widetilde{w}_{n}\right|^{2}-\mu(\beta U) \widetilde{v}_{n}^{2}-2 \mu(\alpha U) \widetilde{v}_{n} \widetilde{w}_{n}-2 \gamma(\beta U)\left|\widetilde{w}_{n}\right|^{2}\right)\right. \\ \left.\quad+P\left(\frac{\varepsilon_{n} x}{\sqrt{P\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right) \widetilde{v}_{n}^{2}+Q\left(\frac{\varepsilon_{n} x}{\sqrt{Q\left(x_{\varepsilon_{n}}\right)}}+x_{\mathcal{E}_{n}}\right) \widetilde{w}_{n}^{2}\right] \\ =\left(P\left(x_{\mathcal{E}_{n}}\right)\right)^{\frac{N}{2}-1}+o_{R}(1)+O\left(e^{-R}\right),\end{array} \end{equation}$$which is impossible for large n and R. So we complete this proof.Proposition 2.5For ε > 0 sufficiently small, there is (φε , ψε) ∈ Eε,xε such thatJε′φε,ψε,(g,h)=0,∀(g,h)∈Eε,xε.$$\begin{equation}\left\langle J_{\varepsilon}^{\prime}\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right),(g, h)\right\rangle=0, \quad \forall(g, h) \in E_{\varepsilon, x_{\varepsilon}}. \end{equation}$$Moreover,φε,ψεε≤C∇Pxε+∇QxεεN2+1+εN2+2$$\begin{equation}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon} \leq C\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right] \end{equation}$$for some constant C > 0 independent of ε.Proof. We will use the contraction mapping theorem to prove the wanted result. By Lemma 2.2, ℓε(φ,ψ)$\ell_{\varepsilon}(\varphi, \psi)$is a bounded linear function in Eε,Xε.$E_{\varepsilon, X_{\varepsilon}}.$So using Riesz representation theorem, we obtain that there is an ℓˉε∈Eε,xε,$\bar{\ell}_{\varepsilon} \in E_{\varepsilon, x_{\varepsilon}},$such thatℓε(φ,ψ)=ℓˉε,(φ,ψ).$$\begin{equation}\ell_{\varepsilon}(\varphi, \psi)=\left\langle\bar{\ell}_{\varepsilon},(\varphi, \psi)\right\rangle. \end{equation}$$Hence finding a critical point for Jε(φ, ψ) is equivalent to solving(2.16)ℓε+L(φ,ψ)+Rε′(φ,ψ)=0.$$\ell_{\varepsilon}+L(\varphi, \psi)+R_{\varepsilon}^{\prime}(\varphi, \psi)=0.$$It follows from Lemma 2.4 that (2.16) can be rewritten as(2.17)(φ,ψ)=A(φ,ψ):=−L−1ℓˉε+Rε′(φ,ψ).$$\begin{equation}(\varphi, \psi)=A(\varphi, \psi):=-L^{-1}\left(\bar{\ell}_{\varepsilon}+R_{\varepsilon}^{\prime}(\varphi, \psi)\right). \end{equation}$$Now we setSε=(φ,ψ)∈Eε,xε,∥(φ,ψ)∥ε≤∇Pxε+∇QxεεN2+1−θ+εN2+2−θ$$\begin{equation}S_{\varepsilon}=\left\{(\varphi, \psi) \in E_{\varepsilon, x_{\varepsilon}},\|(\varphi, \psi)\|_{\varepsilon} \leq\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1-\theta}+\varepsilon^{\frac{N}{2}+2-\theta}\right]\right\} \end{equation}$$with θ ∈ (0, 1). Then for any (φ, ψ) ∈ Sε,∥A(φ,ψ)∥≤Cℓˉεε+Rε′(φ,ψ)≤Cℓˉεε+Cε−N2∥(φ,ψ)∥ε2≤∇Pxε+∇QxεεN2+1−θ+εN2+2−θ.$$\begin{equation}\begin{aligned}\|A(\varphi, \psi)\| & \leq C\left(\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon}+\left\|R_{\varepsilon}^{\prime}(\varphi, \psi)\right\|\right) \\ & \leq C\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon}+C \varepsilon^{-\frac{N}{2}}\|(\varphi, \psi)\|_{\varepsilon}^{2} \\ & \leq\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1-\theta}+\varepsilon^{\frac{N}{2}+2-\theta}\right] . \end{aligned} \end{equation}$$Then A maps Sε to Sε. On the other hand, for any (φ1, ψ1), (φ2, ψ2) ∈ Sε, form Lemma 2.1,Aφ1,ψ1−Aφ2,ψ2=L−1Rε′φ1,ψ1−Rε′φ2,ψ2≤CRε′′tφ1,ψ1+(1−t)φ2,ψ2φ1,ψ1−φ2,ψ2ε≤12φ1,ψ1−φ2,ψ2ε,$$\begin{equation}\begin{aligned}\left\|A\left(\varphi_{1}, \psi_{1}\right)-A\left(\varphi_{2}, \psi_{2}\right)\right\| &=\left\|L^{-1}\left(R_{\varepsilon}^{\prime}\left(\varphi_{1}, \psi_{1}\right)-R_{\varepsilon}^{\prime}\left(\varphi_{2}, \psi_{2}\right)\right)\right\| \\ & \leq C\left\|R_{\varepsilon}^{\prime \prime}\left(t\left(\varphi_{1}, \psi_{1}\right)+(1-t)\left(\varphi_{2}, \psi_{2}\right)\right)\right\|\left\|\left(\varphi_{1}, \psi_{1}\right)-\left(\varphi_{2}, \psi_{2}\right)\right\|_{\varepsilon} \\ & \leq \frac{1}{2}\left\|\left(\varphi_{1}, \psi_{1}\right)-\left(\varphi_{2}, \psi_{2}\right)\right\|_{\varepsilon}, \end{aligned} \end{equation}$$where t ∈ (0, 1). This gives that A is a contraction map from Sε to Sε. Applying the contraction mapping theorem, we can find a unique (φε , ψε) ∈ Sε satisfying (2.17) andφε,ψεε≤∇Pxε+∇QxεεN2+1−θ+εN2+2−θ.$$\begin{equation}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon} \leq\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1-\theta}+\varepsilon^{\frac{N}{2}+2-\theta}\right] . \end{equation}$$Furthermore, in view of (2.17), we getφε,ψεε=L−1ℓˉε+L−1Rε′φε,ψεε≤Cℓˉεε+Cε−N2φε,ψεε2≤Cℓˉεε+C∇Pxε+∇QxEε1−θ+ε2−θφε,ψεε,$$\begin{equation}\begin{aligned}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon} &=\left\|L^{-1}\left(\bar{\ell}_{\varepsilon}\right)+L^{-1}\left(R_{\varepsilon}^{\prime}\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right)\right\|_{\varepsilon} \\ & \leq C\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon}+C \varepsilon^{-\frac{N}{2}}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon}^{2} \\ & \leq C\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon}+C\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\mathcal{E}}\right)\right|\right) \varepsilon^{1-\theta}+\varepsilon^{2-\theta}\right]\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon}, \end{aligned} \end{equation}$$which implies thatφε,ψεε≤Cℓˉεε≤C∇Pxε+∇QxεεN2+1+εN2+2.$$\begin{equation}\left\|\left(\varphi_{\varepsilon}, \psi_{\varepsilon}\right)\right\|_{\varepsilon} \leq C\left\|\bar{\ell}_{\varepsilon}\right\|_{\varepsilon} \leq C\left[\left(\left|\nabla P\left(x_{\varepsilon}\right)\right|+\left|\nabla Q\left(x_{\varepsilon}\right)\right|\right) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right]. \end{equation}$$3Proof of our main resultIn this section, we assume that x0 ∈ Γ is a non-degenerate critical point of Δ(P + Q) and we will construct a single peak solution (vε , wε) of (1.1) concentrating at x0.From Proposition 2.5, we can get the following result.Proposition 3.1There exists an ε0 > 0such that for any ε ∈ (0, ε0] and y close to x0, there is (φε,y , ψε,y) ∈ Eε,y such that for any (g, h) ∈ Eε,y,∫ε2∇Vε,y+φε,y∇g+P(x)Vε,y+φε,yg+ε2∇Wε,y+ψε,y∇h+Q(x)Wε,y+ψε,yh−∫μVε,y+φε,ygWε,y+ψε,y+μ2Vε,y+φε,y2h+γWε,y+ψε,y2h=0.$$\begin{equation}\begin{array}{l}\int\left[\varepsilon^{2} \nabla\left(V_{\varepsilon, y}+\varphi_{\varepsilon, y}\right) \nabla g+P(x)\left(V_{\varepsilon, y}+\varphi_{\varepsilon, y}\right) g+\varepsilon^{2} \nabla\left(W_{\varepsilon, y}+\psi_{\varepsilon, y}\right) \nabla h\right. \\ \left.+Q(x)\left(W_{\varepsilon, y}+\psi_{\varepsilon, y}\right) h\right]-\int\left[\mu\left(V_{\varepsilon, y}+\varphi_{\varepsilon, y}\right) g\left(W_{\varepsilon, y}+\psi_{\varepsilon, y}\right)+\frac{\mu}{2}\left(V_{\varepsilon, y}+\varphi_{\varepsilon, y}\right)^{2} h\right. \\ \left.+\gamma\left(W_{\varepsilon, y}+\psi_{\varepsilon, y}\right)^{2} h\right]=0.\end{array} \end{equation}$$Moreover,φε,y,ψε,yε=O(|∇P(y)|+|∇Q(y)|)εN2+1+εN2+2.$$\begin{equation}\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}=O\left((|\nabla P(y)|+|\nabla Q(y)|) \varepsilon^{\frac{N}{2}+1}+\varepsilon^{\frac{N}{2}+2}\right) . \end{equation}$$To get a true solution of (1.1), we need to choose y such that(3.1)∫ε2∇vε∇∂vε∂xi+P(x)vε∂vε∂xi+ε2∇wε∇∂wε∂xi+Q(x)wε∂wε∂xi−∫μvε∂vε∂xiwε+μ2vε2∂wε∂xi+γwε2∂wε∂xi=0,$$\begin{equation}\begin{array}{c}\int\left[\varepsilon^{2} \nabla v_{\varepsilon} \nabla \frac{\partial v_{\varepsilon}}{\partial x_{i}}+P(x) v_{\varepsilon} \frac{\partial v_{\varepsilon}}{\partial x_{i}}+\varepsilon^{2} \nabla w_{\varepsilon} \nabla \frac{\partial w_{\varepsilon}}{\partial x_{i}}+Q(x) w_{\varepsilon} \frac{\partial w_{\varepsilon}}{\partial x_{i}}\right]-\int\left[\mu v_{\varepsilon} \frac{\partial v_{\varepsilon}}{\partial x_{i}} w_{\varepsilon}\right. \\ \left.+\frac{\mu}{2} v_{\varepsilon}^{2} \frac{\partial w_{\varepsilon}}{\partial x_{i}}+\gamma w_{\varepsilon}^{2} \frac{\partial w_{\varepsilon}}{\partial x_{i}}\right]=0,\end{array} \end{equation}$$where vε=Vε,y+φε,y,wε=Wε,y+ψε,y$v_{\varepsilon}=V_{\varepsilon, y}+\varphi_{\varepsilon, y}, w_{\varepsilon}=W_{\varepsilon, y}+\psi_{\varepsilon, y}$and i = 1, · · · , N. It is easy to check that (3.1) is equivalent to(3.2)∫∂P(x)∂xivε2+∂Q(x)∂xiwε2=0,i=1,⋯,N,$$\begin{equation}\int\left[\frac{\partial P(x)}{\partial x_{i}} v_{\varepsilon}^{2}+\frac{\partial Q(x)}{\partial x_{i}} w_{\varepsilon}^{2}\right]=0, i=1, \cdots, N, \end{equation}$$which is exact the Pohozaev type identity.For y close to x0, y ∈ Γt for some t close to 1. In the following, we denote by ν the unit normal vector of Γt at y and we use ςi , i = 1, · · · , N − 1, to denote the principal direction of Γt at y. Then, at y, one hasDζiP(y)=0,|∇P(y)|=DvP(y)$$\begin{equation}D_{\zeta_{i}} P(y)=0, \quad|\nabla P(y)|=\left|D_{v} P(y)\right| \end{equation}$$andDζiQ(y)=0,|∇Q(y)|=DvQ(y).$$\begin{equation}D_{\zeta_{i}} Q(y)=0,|\nabla Q(y)|=\left|D_{v} Q(y)\right|. \end{equation}$$First, we prove the following results.Lemma 3.2If (H1)− (H3) hold, then∫DvP(x)vε2+DvQ(x)wε2=0$$\begin{equation}\int\left(D_{v} P(x) v_{\varepsilon}^{2}+D_{v} Q(x) w_{\varepsilon}^{2}\right)=0 \end{equation}$$is equivalent toDvP(y)+DvQ(y)=Oε2.$$\begin{equation}D_{v} P(y)+D_{v} Q(y)=O\left(\varepsilon^{2}\right). \end{equation}$$Proof. By the direct computations, we have(3.3)∫DvP(x)Vε,y2+DvQ(x)Wε,y2=−∫2DvP(x)Vε,yφε,y+DvP(x)φε,y2+2DvQ(x)Wε,yψε,y+DvQ(x)ψε,y2=ODVP(y)+DvQ(y)εN2φε,y,ψε,yε+εN2+1φε,y,ψε,yε+φε,y,ψε,yε2=εNODvP(y)+DνQ(y)ε+ε2.$$\begin{equation}\begin{array}{l}\int\left(D_{v} P(x) V_{\varepsilon, y}^{2}+D_{v} Q(x) W_{\varepsilon, y}^{2}\right) \\ =-\int\left[2 D_{v} P(x) V_{\varepsilon, y} \varphi_{\varepsilon, y}+D_{v} P(x) \varphi_{\varepsilon, y}^{2}+2 D_{v} Q(x) W_{\varepsilon, y} \psi_{\varepsilon, y}+D_{v} Q(x) \psi_{\varepsilon, y}^{2}\right] \\ =O\left(\left(\left|D_{V} P(y)\right|+\left|D_{v} Q(y)\right|\right) \varepsilon^{\frac{N}{2}}\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}+\varepsilon^{\frac{N}{2}+1}\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}+\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}^{2}\right) \\ =\varepsilon^{N} O\left(\left(\left|D_{v} P(y)\right|+\left|D_{\nu} Q(y)\right|\right) \varepsilon+\varepsilon^{2}\right) .\end{array} \end{equation}$$On the other hand, using Taylor’s expansion, we get(3.4)∫DvP(x)Vε,y2+DvQ(x)Wε,y2=εN∫Bδε(0)DvP(y)+ε2|X|22!ΔDvP(y)α2P2(y)U2(P(y)x)+∫Bδε(0)DvQ(y)+ε2|X|22!ΔDvQ(y)β2Q2(y)U2(Q(y)x)+OεN+4=ODvP(y)+DvQ(y)εN+ΔDvP(y)+ΔDvQ(y)εN+2+εN+4.$$\begin{equation}\begin{array}{l}\int\left(D_{v} P(x) V_{\varepsilon, y}^{2}+D_{v} Q(x) W_{\varepsilon, y}^{2}\right) \\ =\varepsilon^{N}\left[\int_{B_{\frac{\delta}{\varepsilon}}(0)}\left(D_{v} P(y)+\frac{\varepsilon^{2}|X|^{2}}{2 !}\left(\Delta D_{v} P\right)(y)\right) \alpha^{2} P^{2}(y) U^{2}(\sqrt{P(y)} x)\right. \\ \left.+\int_{B_{\frac{\delta}{\varepsilon}}(0)}\left(D_{v} Q(y)+\frac{\varepsilon^{2}|X|^{2}}{2 !}\left(\Delta D_{v} Q\right)(y)\right) \beta^{2} Q^{2}(y) U^{2}(\sqrt{Q(y)} x)\right]+O\left(\varepsilon^{N+4}\right) \\ =O\left(\left(\left|D_{v} P(y)\right|+\left|D_{v} Q(y)\right|\right) \varepsilon^{N}+\left(\left(\Delta D_{v} P\right)(y)+\left(\Delta D_{v} Q\right)(y)\right) \varepsilon^{N+2}+\varepsilon^{N+4}\right).\end{array} \end{equation}$$Combining (3.3) and (3.4), we findDvP(y)+DvQ(y)=Oε2.$$\begin{equation}D_{v} P(y)+D_{v} Q(y)=O\left(\varepsilon^{2}\right). \end{equation}$$Lemma 3.3Under the conditions (H1)− (H3),∫DζP(x)vε2+DζQ(x)wε2=0$$\begin{equation}\int\left(D_{\zeta} P(x) v_{\varepsilon}^{2}+D_{\zeta} Q(x) w_{\varepsilon}^{2}\right)=0 \end{equation}$$is equivalent toDςΔP(y)+DςΔQ(y)+B1K(y)ε2=O∇G1(y)+∇G2(y)(|∇P(y)|+|∇Q(y)|)+∇G1(y)+∇G2(y)ε,$$\begin{equation}\begin{array}{l}\left(D_{\varsigma} \Delta P\right)(y)+\left(D_{\varsigma} \Delta Q\right)(y)+B_{1} K(y) \varepsilon^{2} \\ =O\left(\left(\left|\nabla G_{1}(y)\right|+\left|\nabla G_{2}(y)\right|\right)(|\nabla P(y)|+|\nabla Q(y)|)+\left(\left|\nabla G_{1}(y)\right|+\left|\nabla G_{2}(y)\right|\right) \varepsilon\right),\end{array} \end{equation}$$where B1 is some constant, K(y) is a smooth function, and G1(x) = ∇P(x), ς, G2(x) = ∇Q(x), ς.Proof. Since for any fixed d > 0 and j = 1,2Gj(x)=∑i=1N∂Gj(y)∂yixi−yi+12∑i=1N∑ℓ=1N∂2Gj(y)∂yi∂yℓxi−yixℓ−yℓ+o|x−y|2, in Bd(y),$$\begin{equation}G_{j}(x)=\sum_{i=1}^{N} \frac{\partial G_{j}(y)}{\partial y_{i}}\left(x_{i}-y_{i}\right)+\frac{1}{2} \sum_{i=1}^{N} \sum_{\ell=1}^{N} \frac{\partial^{2} G_{j}(y)}{\partial y_{i} \partial y_{\ell}}\left(x_{i}-y_{i}\right)\left(x_{\ell}-y_{\ell}\right)+o\left(|x-y|^{2}\right), \text { in } B_{d}(y), \end{equation}$$we have∫G1(x)Vε,y2+G2(x)Wε,y2=−2∫G1(x)Vε,yφε,y−∫G1(x)φε,y2−2∫G2(x)Wε,yψε,y−∫G2(x)ψε,y2=OεN2ε∇G1(y)+ε∇G2(y)+ε2φε,y,ψε,yε+ε∇G1(y)+∇G2(y)φε,y,ψε,yε2=O∇G1(y)+∇G2(y)(|∇P(y)|+|∇Q(y)|)εN+2+∇G1(y)+∇G2(y)εN+3+εN+4.$$\begin{equation}\begin{array}{l}\int\left(G_{1}(x) V_{\varepsilon, y}^{2}+G_{2}(x) W_{\varepsilon, y}^{2}\right) \\ =-2 \int G_{1}(x) V_{\varepsilon, y} \varphi_{\varepsilon, y}-\int G_{1}(x) \varphi_{\varepsilon, y}^{2}-2 \int G_{2}(x) W_{\varepsilon, y} \psi_{\varepsilon, y}-\int G_{2}(x) \psi_{\varepsilon, y}^{2} \\ =O\left(\varepsilon^{\frac{N}{2}}\left(\varepsilon\left|\nabla G_{1}(y)\right|+\varepsilon\left|\nabla G_{2}(y)\right|+\varepsilon^{2}\right)\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}+\varepsilon\left(\left|\nabla G_{1}(y)\right|\right.\right. \\ \left.\left.\quad+\left|\nabla G_{2}(y)\right|\right)\left\|\left(\varphi_{\varepsilon, y}, \psi_{\varepsilon, y}\right)\right\|_{\varepsilon}^{2}\right) \\ =O\left(\left(\left|\nabla G_{1}(y)\right|+\left|\nabla G_{2}(y)\right|\right)(|\nabla P(y)|+|\nabla Q(y)|) \varepsilon^{N+2}+\left(\left|\nabla G_{1}(y)\right|+\left|\nabla G_{2}(y)\right|\right) \varepsilon^{N+3}\right. \\ \left.\quad+\varepsilon^{N+4}\right).\end{array} \end{equation}$$On the other hand, from Gj(y) = 0, j = 1,2, we get∫G1(x)Vε,y2+G2(x)Wε,y2=OεN+2ΔG1(y)+ΔG2(y)+B1K(y)εN+4+εN+6.$$\begin{equation}\begin{array}{l}\int\left(G_{1}(x) V_{\varepsilon, y}^{2}+G_{2}(x) W_{\varepsilon, y}^{2}\right) \\ =O\left(\varepsilon^{N+2}\left(\Delta G_{1}(y)+\Delta G_{2}(y)\right)+B_{1} K(y) \varepsilon^{N+4}+\varepsilon^{N+6}\right).\end{array} \end{equation}$$As a result, the result follows.Proof of Theorem 1.4Now, by Lemmas 3.2 and 3.3, (3.2) is equivalent to(3.5)DvP(y)+DvQ(y)=Oε2,DςΔP(y)+DςΔQ(y)=O|∇P(y)|+|∇Q(y)|+ε2,$$\begin{equation}D_{v} P(y)+D_{v} Q(y)=O\left(\varepsilon^{2}\right),\left(D_{\varsigma} \Delta P\right)(y)+\left(D_{\varsigma} \Delta Q\right)(y)=O\left(|\nabla P(y)|+|\nabla Q(y)|+\varepsilon^{2}\right), \end{equation}$$which is also equivalent to(3.6)DvP(y)+DvQ(y)=Oε2,DςΔP(y)+DςΔQ(y)=Oε2.$$\begin{equation}D_{v} P(y)+D_{v} Q(y)=O\left(\varepsilon^{2}\right),\left(D_{\varsigma} \Delta P\right)(y)+\left(D_{\varsigma} \Delta Q\right)(y)=O\left(\varepsilon^{2}\right) . \end{equation}$$Let yˉ$\bar{y}$∈ Γ be the point such that y−yˉ=κv$y-\bar{y}=\kappa v$for some κ ∈ ℝ. We have DvP(yˉ)=0 and DvQ(yˉ)=0.$D_{v} P(\bar{y})=0 \text { and } D_{v} Q(\bar{y})=0.$As a result,DvP(y)+DvQ(y)=DvP(y)−DvP(yˉ)+DvQ(y)−DvQ(yˉ)=Dvv2P(yˉ)+Dvv2Q(yˉ)〈y−yˉ,v〉+O|y−yˉ|2,$$\begin{equation}\begin{aligned} D_{v} P(y)+D_{v} Q(y) &=D_{v} P(y)-D_{v} P(\bar{y})+D_{v} Q(y)-D_{v} Q(\bar{y}) \\ &=\left(D_{v v}^{2} P(\bar{y})+D_{v v}^{2} Q(\bar{y})\right)\langle y-\bar{y}, v\rangle+O\left(|y-\bar{y}|^{2}\right), \end{aligned} \end{equation}$$which, together with the non-degenerate assumption, yields that DνP(y) + DνQ(y) = O(ε2) can be written as(3.7)〈y−yˉ,v〉=Oε2+|y−yˉ|2.$$\begin{equation}\langle y-\bar{y}, v\rangle=O\left(\varepsilon^{2}+|y-\bar{y}|^{2}\right). \end{equation}$$Let ςˉi$\bar{\varsigma}_i$be the i − th tangential unit vector of Γ at yˉ.$\bar{y}.$It follows from the assumption (H2) that(DςiΔP)(y)+(DςiΔQ)(y)=(DςˉiΔP)(yˉ)+(DςˉiΔQ)(yˉ)+O(|y−yˉ|)=(DςˉiΔP)(yˉ)+(DςˉiΔQ)(yˉ)+O(ε2)$$\begin{aligned}(D_{\varsigma_i} \Delta P)(y)+(D_{\varsigma_i} \Delta Q)(y)&= (D_{\bar{\varsigma}_i} \Delta P)(\bar{y})+(D_{\bar{\varsigma}_i} \Delta Q)(\bar{y})+O(|y-\bar{y}|)\\ &=(D_{\bar{\varsigma}_i} \Delta P)(\bar{y})+(D_{\bar{\varsigma}_i} \Delta Q)(\bar{y})+O(\varepsilon^2) \end{aligned}$$and(DςˉiΔP)(yˉ)+(DςˉiΔQ)(yˉ)=(DςˉiΔP)(yˉ)−(Dςˉi,0ΔP)(x0)+(DςˉiΔQ)(yˉ)−(Dςˉi,0ΔQ)(x0)=〈(∇TDςi,0ΔP)(x0),yˉ−x0〉+〈(∇TDςi,0ΔQ)(x0),yˉ−x0〉+O(|yˉ−x0|2),$$\begin{aligned}&(D_{\bar{\varsigma}_i} \Delta P)(\bar{y})+(D_{\bar{\varsigma}_i} \Delta Q)(\bar{y})\\ &=(D_{\bar{\varsigma}_i} \Delta P)(\bar{y})-(D_{\bar{\varsigma}_{i,0}} \Delta P)(x_0)+ (D_{\bar{\varsigma}_i} \Delta Q)(\bar{y})-(D_{\bar{\varsigma}_{i,0}} \Delta Q)(x_0)\\ &=\langle (\nabla_T D_{\varsigma_{i,0}}\Delta P)(x_0),\bar{y}-x_0\rangle+\langle (\nabla_T D_{\varsigma_{i,0}}\Delta Q)(x_0),\bar{y}-x_0\rangle +O(|\bar{y}-x_0|^2), \end{aligned}$$where ∇T is the tangential gradient on Γ at x0 and ςi,0 is the i − th tangential unit vector of Γ at x0. Hence (DςΔP)(y) + (DςΔQ)(y) = O(ε2) can be rewritten as(3.8)〈(∇TDςi,0ΔP)(x0)+(∇TDςi,0ΔQ)(x0),yˉ−x0〉=O(ε2+|yˉ−x0|2).$$\begin{aligned}\langle (\nabla_T D_{\varsigma_{i,0}}\Delta P)(x_0)+ (\nabla_T D_{\varsigma_{i,0}}\Delta Q)(x_0),\bar{y}-x_0\rangle =O(\varepsilon^2+|\bar{y}-x_0|^2). \end{aligned}$$So we can solve (3.7) and (3.8) to get y = xε with xε → x0 as ε → 0.

Journal

Advances in Nonlinear Analysis – de Gruyter

Published: Jan 1, 2022

Keywords: second-harmonic generation; non-degenerate; single peak solutions; reduction method; 35J10; 35B99; 35J60

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