Existence of renormalized solutions for nonlinear elliptic problem in Musielak-Orlicz-Sobolev spaces

Abstract

In this paper, we prove the existence of renormalized solutions for some class nonlinear elliptic problem of the type

−div a(x,u,∇u) + H(x,u,∇u) = µ − div φ(u),

in the Musielak-Orlicz-Sobolev spaces W¹₀Lϕ(Ω). No ∆₂−condition is assumed on the Musielak function. We assume that H(x,s,ξ) satisfies has a natural growth with respect to its third argument and satisfies the sign condition. The µ is assumed to belong to L¹(Ω) + W⁻¹Eψ(Ω) and φ(·) ∈ C⁰(IR,IR^N) is a continuous function.