Renormalized solutions for some non-coercive quasilinear elliptic problem in Musielak-Orlicz space

Abstract

In this paper, we study the existence of renormalized solutions for the following non-coercive quasilinear elliptic problem: −div(a(x,u,∇u))+g(x,u) =f−div(φ(u)) in Ω, and u=0 on ∂Ω. in the Musielak-Orlicz-Sobolev space W¹₀L_φ(Ω), where−diva(x,u,∇u) is a degenerate Leary Lions operator and g(x,u) is a Caratheodory function that satisfies the sign condition with φ(·) ∈ C₀(R,R^N) and f ∈ L¹(Ω).The Musielak-Orlicz function φ(x,t) is regular and does not necessarily satisfying the ∆₂−condition.