On a quasilinear Poisson equation in the plane

Abstract

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