Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems

Abstract

In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion $\beta(u)-div(a(x,Du)+F(u))\ni f$ in $\Omega $, where $f\in L^{1}(\Omega )$. A vector field $a(\cdot,\cdot)$ is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general $L^1$-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.