Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data

Abstract

We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type: $$- div(a(x,u,\nabla u) + \varphi (u)) + g(x,u,\nabla u) = \mu ,$$ where the right-hand side belongs to L 1(Ω) + W −1,p′(x)(Ω), -div(a(x, u,∇u)) is a Leray-Lions operator defined from W −1,p′(x)(Ω) into its dual and φ ∈ C 0(ℝ,ℝ N ). The function g(x, u,∇u) is a non linear lower order term with natural growth with respect to |∇u| satisfying the sign condition, that is, g(x, u,∇u)u ≥ 0.