Existence result of entropy solution for degenerate elliptic problem without monotonicity condition in Generalized Sobolev spaces

In the present paper we prove some existence results of entropy solution for nonlinear degenerate elliptic problems of the form $A u+h(x, u)=f,$ in Musielak-Orlicz-Sobolev spaces, where A(u)=-div (a(x,u,∇ u)) is a Leray-Lions, operator defined form the musielak-Orlicz-sobolev spaces W₀¹Lφ(Ω) into its dual.The right hand side f ∈ L¹(Ω), and no monotonicity strict condition is assumed on the function a(x, s, ξ), The tool we use to overcome this difficulty is to investigate some techniques introduced by Minty's lemma.