Existence Result for Some Nonlinear Lφ-Elliptic Problems

Abstract

The present paper is dedicated to study the existence of renormalized solutions for the following elliptic problem beginequationlabelmypb begincases displaystyle -divleft(a(x, u, ∇ u)right)+H(x, u, ∇ u)=μ-divleft(φ(x, u)right)-divleft(Φ(u)right) & in Ω, u=0 & on ∂ Ω, endcases endequation in the setting of Musielak-Orlicz Sobolev spaces, where the Caratheodory function φ(x, u) satisfies certain growth conditions, H(x,s,ξ) satisfies the sign and the growth condition and Φ ∈ C⁰(ℝ,mathbbRᴺ), the datum μ= f-div(F) is a measure with f∈ L¹(Ω) and F∈ (Eφ(Ω))ᴺ.