Existence of Solution for a General Class of Strongly Nonlinear Elliptic Problems Having Natural Growth Terms and $L^1$ -Data

Abstract

This paper is concerned with the existence of solution for a general class of strongly nonlinear elliptic problems associated with the differential inclusion $$β(u)+A(u)+g(x,u,Du) \ni f,$$ where $A$ is a Leray-Lions operator from $W^{1,p}_0(Ω)$ into its dual, $β$ maximal monotone mapping such that $0 ∈ β(0),$ while $g(x,s, ξ)$ is a nonlinear term which has a growth condition with respect to $ξ$ and no growth with respect to $s$ but it satisfies a sign-condition on $s.$ The right hand side $f$ is assumed to belong to $L^1(Ω).$