Abstract
Our aim in this paper is to study the existence of renormalized solution for a class of nonlinear p(x)-Laplace problems with Neumann nonhomogeneous boundary conditions and diuse Radon measure data which does not charge the sets of zero p(.)-capacity
Highlights
The operator −∆p(x)u is called p(x)-Laplacian which become p-Laplacian when p(x) ≡ p is a constant
The interest of study problem involving variable exponent is due to the fact that they can model various phenomena which arise in the study of elastic mechanics, electrorheological fluids or image restauration
This notion was introduced by DiPerna and Lions in [12] for the first order equations and has been developed for elliptic problems with Dirichlet boundary conditions and with L1(Ω) data in [17]
Summary
As the exponent p(x) appearing in (Pμ) depends on the variable x, we must work with Lebesgue and Sobolev spaces with variable exponents, under the following assumptions on the data: p(.) : Ω → IR is a continuous function such that 1 < p− ≤ p+ < +∞,. We define the Lebesgue space with variable exponent Lp(.)(Ω) as the set of all measurable functions u : Ω → IR for which the convex modular ρp(x)(u) := |u|p(x)dx is finite. Let W 1,p(x)(Ω) = {u ∈ Lp(x)(Ω) and |∇u| ∈ Lp(x)(Ω)}, which is a Banach space equipped with the following norm (2.2). For a measurable function u : Ω → IR, we introduce the following notation: ρ1,p(x)(u) = |u|p(x) dx + |∇u|p(x) dx. Let us introduce the following notation: Given two bounded measurable functions p(x), q(x) : Ω → IR, we write q(x) ≪ p(x) if ess inf (p(x) − q(x)) > 0. Let (vn)n∈IN be a sequence of measurable functions in Ω. Let (X, M, μ) be a measure space such that μ(X) < +∞. Μ ∈ Mpb(.)(X) if and only if μ ∈ L1(X) + W −1,p′ (.)(X)
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