Existence of entropy solutions of the anisotropic elliptic nonlinear problem with measure data in weighted Sobolev space

Abstract

This paper is devoted to study the following nonlinear anisotropic elliptic unilateral problem \begin{equation*} \begin{cases} A\,u -\mbox{div}\,\phi(u)=\mu \quad \mbox{in} \qquad \Omega \\ \;u=0 \qquad \mbox{on} \quad \partial \Omega , \end{cases} \end{equation*} where the right hand side $\,\mu\;$ belongs to $\; L^1(\Omega)+ W_{0}^{-1,\overrightarrow{p}'} (\Omega,\ \overrightarrow{\omega}^*)$. The operator $\displaystyle A\,u=-\sum_{i=1}^{N}\partial_{i}\,a_{i}(x,\ u,\ \nabla u)$ is a Leray-Lions anisotropic operator acting from $\; W_{0}^{1,\overrightarrow{p}} (\Omega,\ \overrightarrow{\omega})\;$ into its dual $\; W_{0}^{-1,\overrightarrow{p}'} (\Omega,\ \overrightarrow{\omega}^*)$ and $\phi_{i}\in C^{0}(\mathbb{R},\mathbb{R})$.