Renormalized Solutions for Some Nonlinear Degenerated Elliptic Problems in Weighted Sobolev Space with Variable Exponents

Abstract

In this article, we prove the existence of a renormalised solution to the problem of the nonlinear elliptic equation:
\begin{equation*}
\left\{
\begin{array}{c}
-\func{div}\left( a(x,u,Du)\right) =f\text{ \ in }\Omega , \\
u=0\text{ on }\partial \Omega.%
\end{array}%
\right \label{ros}
\end{equation*}
Where $
a(x,u,Du)$ is below-up for a finite value $m$ of the unknown $u$ and the data $ f\in L^{1}(\Omega) $.}