Existence and uniqueness of a solutions of nonlinear degenerated equations with measure data

Abstract

This paper is devoted to the study of the following of nonlinear degenerate equations of the type \begin{eqnarray*} (P)\quad \left\{\begin{array}{rl} Au &=\>\mu \quad \mbox{in} \quad \Omega \\ u \>& =\>  0 \quad \mbox{on} \quad \partial\Omega, \end{array}\right. \end{eqnarray*} where $A$ is a Leray Lions operator acted from weighted Sobolev space $W_{0}^{1, p}(\Omega,\omega)$ into its dual $W^{-1, p'}(\Omega,\omega^{\ast})$, and $\mu$  is a Radon measure does not charge the sets  of null $(p,\omega)$-capacity. We prove a decomposition theorem for measure does not charge the sets  of null $(p,\omega)$-capacity. We apply this result to prove existence and uniqueness of an entropy solution of problem $(P)$.