Existence results for non-uniformly elliptic equations with general growth in the gradient

Abstract

We prove a priori estimates and existence results for a class of problems whose prototype is \[ -\text{div}\,(b( |{u}| ) | {Du}| ^{p-2}Du)=k( |{u} | ) |{Du}|^q+f,\quad u\in W_0^{1,p}(\Omega)\cap L^\infty(\Omega), \] where $\Omega$ is a bounded domain in $\mathbb R^n$, $p-1 < q\leq p$, and $k$ and $b$ are continuous functions.