Existence results for Dirichlet problems with degenerated p-Laplacian

Abstract

In this article, we prove the existence of entropy solutions for the Dirichlet problem $(P)\left\{ \begin{array}{ll} & -{\rm div}[{\omega}(x){\vert{\nabla}u\vert}^{p-2}{\nabla}u]= f(x) - {\rm div}(G(x)),\ \ {\rm in} \ \ {\Omega} \\ & u(x)=0, \ \ {\rm in} \ \ {\partial\Omega} \end{array} \right.$ where $\Omega$ is a bounded open set of $\mathbb{R}^N$ $(N \geq 2)$, $f \in L^1(\Omega)$ and $G/\omega \in [L^p(\Omega,\omega)]^N$.