Existence of Renormalized Solutions of Nonlinear Elliptic Problems in Weighted Variable-Exponent Space

In this article, we study a general class of nonlinear degenerated ellipticproblems associated with the differential inclusion $β(u)-div(a(x,Du)+F(u)) ∋ f$ in $Ω$ where $f ∈ L^1(Ω).$ A vector field $a(.,.)$ is a Carathéodory function. Using truncationtechniques and the generalized monotonicity method in the framework of weightedvariable exponent Sobolev spaces, we prove existence of renormalized solutions forgeneral $L^1$-data.