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 Hölder continuous of some order $\beta \in (0,1)$, then u is Hölder continuous of the same order in $\overline{{\mathbb {D}}}$. We extend these results to arbitrary smooth ($C^1$) domains.