We study the Dirichlet problem for the quasilinear partial differential equation $$\triangle u(z) = h(z)\cdot f(u(z))$$ ▵ u ( z ) = h ( z ) · f ( u ( z ) ) in the unit disk $${\mathbb {D}}\subset {\mathbb {C}}$$ D ⊂ C with arbitrary continuous boundary data $$\varphi :\partial {\mathbb {D}}\rightarrow {\mathbb {R}}$$ φ : ∂ D → R . The multiplier $$h:{\mathbb {D}}\rightarrow {\mathbb {R}}$$ h : D → R is assumed to be in the class $$L^p({\mathbb {D}}),$$ L p ( D ) , $$p>1,$$ p > 1 , and the continuous function $$f:\mathbb {R}\rightarrow {\mathbb {R}}$$ f : R → R is such that $$f(t)/t\rightarrow 0$$ f ( t ) / t → 0 as $$t\rightarrow \infty .$$ t → ∞ . Applying the potential theory and the Leray–Schauder approach, we prove the existence of continuous solutions u of the problem in the Sobolev class $$W^{2,p}_{\mathrm{loc}}({\mathbb {D}})$$ W loc 2 , p ( D ) . Furthermore, we show that $$u\in C^{1,\alpha }_{\mathrm{loc}}({\mathbb {D}})$$ u ∈ C loc 1 , α ( D ) with $$\alpha = (p-2)/p$$ α = ( p - 2 ) / p if $$p>2$$ p > 2 and, in particular, with arbitrary $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) if the multiplier h is essentially bounded. In the latter case, if in addition $$\varphi $$ φ is Hölder continuous of some order $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) , then u is Hölder continuous of the same order in $$\overline{{\mathbb {D}}}$$ D ¯ . We extend these results to arbitrary smooth ( $$C^1$$ C 1 ) domains.
Analysis and Mathematical Physics – Springer Journals
Published: Jan 1, 2020
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