Entropy solutions for nonlinear nonhomogeneous Neumann problems involving the generalized p(x)-Laplace operator

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Résumé

The paper deals with the inhomogeneous nonlinear Neumann boundary value problem
−div(Φ(∇u−Θ(u)))+|u|p(x)−2u+α(u)=finΩ,

(Φ(∇u−Θ(u))⋅η+γ(u)=gonΩ
with
Φ(ξ)=|ξ|p(x)−2ξ,∀ξ∈ℝN,
where Ω⊆ℝN (N≥3) is a bounded open domain with Lipschitz boundary ∂Ω, η is the outer unit normal vector on ∂Ω, α, γ, Θ are real functions defined on ℝ of ℝN, f∈L1(Ω), g∈L1(∂Ω) and p:Ω¯→ℝ is a continuous function such that 1<p−≤p+<+∞, p−=ess infx∈Ωp(x) and p+=ess supx∈Ωp(x). The techniques of entropy solutions for elliptic equations are used to prove the existence of at least one entropy solution.

Mots-clés

generalized Sobolev spaces; Neumann boundary conditions; entropy solution; noncoercive operator