Strongly Nonlinear $p(x)$-Elliptic Problems with $L^1$-Data

Abstract

In this paper, we will study the existence of solutions in the sense of distributions for the quasilinear $p(x)$-elliptic problem, $$ Au + g(x,u,\nabla u) = f,$$ where $A$ is a Leray-Lions operator from $W_{0}^{1,p(\cdot)}(\Omega)$ into its dual, the nonlinear term $g(x,s,\xi)$ has a growth condition with respect to $\xi$ and the sign condition with respect to $s.$ The datum $\>f\>$ is assumed in the dual space $\>W^{-1,p'(\cdot)}(\Omega),\>$ and then in $\>L^{1}(\Omega).$