Existence of entropy solutions for some nonlinear elliptic problems involving variable exponent and measure data

Abstract

In this paper, we study the existence of entropy solutions for some nonlinear $p(x)-$elliptic equation of the type $$Au - \mbox{div }\phi(u) + H(x,u,\nabla u) = \mu,$$ where $A$ is an operator of Leray-Lions type acting from $W_{0}^{1,p(x)}(\Omega)$ into its dual, the strongly nonlinear term $H$ is assumed only to satisfy some nonstandard growth condition with respect to $|\nabla u|,$ here $\>\phi(\cdot)\in C^{0}(I\!\!R,I\!\!R^{N})\>$ and $\mu$ belongs to ${\mathcal{M}}_{0}^{b}(\Omega)$.