Existence of solutions for a class of degenerate elliptic equations in $P(x)$-Sobolev spaces

Abstract

We study the Dirichlet problem for degenerate elliptic equations of the form \begin{equation*} - \mbox{div} \, a(x,u,\nabla u)+ H(x,u,\nabla u)= f \quad \mbox{in } \Omega, \end{equation*} where $ a(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown $u$, and $H(x,u,\nabla u)$ is a nonlinear term without sign condition. Under suitable conditions on $a$ and $H$, we prove the existence of bounded and unbounded solution for a datum $f\in L^m$, with $1\leq m\leq \infty$.