Existence and regularity for a class of non-uniformly elliptic equations in two dimensions

Abstract

We prove some existence and regularity results for solutions of equations in the form $ -\mathrm{div}(a(x,u) \nabla u) = f$, where $a(x,s) : \Omega \times {\mathbb R} \rightarrow {\mathbb R}$ is a bounded Carath\'eodory function satisfying the inequality $a(x,s)\ge (1+|s|)^{-\theta}$ with $0 \leq \theta \leq1$ and $\Omega$ is a bounded open set of ${\mathbb R}^2$.