Existence of solution for nonlinear elliptic inclusion problems with degenerate coercivity and $L^{1}$-data

Abstract

This paper is concerned with the study of the existence results to the general class of nonlinear elliptic problem associated with the differential inclusion having degenerate coercivity, whose prototype is giving by:

$$\begin{aligned} {\left\{ \begin{array}{ll} \beta (u)-div \,\left( \dfrac{|\nabla u|^{p-2} \nabla u }{(1+|u|)^{\theta (p-1)}}\right) \ni f \, \, &{}\text {in}\ \Omega , \\ u=0 \, \, \, &{} \text {on} \, \partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega$ is a bounded domain in ${\mathbb {R}}^N (N\ge 2)$ with sufficiently smooth boundary $\partial \Omega$, $0 \le \theta < 1$, $1<p<N$, $\beta$ is a maximal monotone mapping such that $0 \in \beta (0)$ and the right hand side f is assumed to belong to $L^1(\Omega )$. We show the existence of entropy solutions for this non-coercive differential inclusion and we will conclude some regularity results.