Entropy solutions for some elliptic anisotropic problems involving variable exponent with Fourier boundary conditions and measure data

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Résumé

This paper is devoted to the study of some nonlinear elliptic anisotropic Fourier boundary-value problems, whose prototype is given by

{−Au+g(x,u,∇u)+δ∣u∣p0(x)−2u=μ−div⁡ϕ(u)in Ω,Bu+λu=hon ∂Ω,\begin{cases} -Au+g(x,u,\nabla u)+\delta |u|^{p_0(x)-2}u =\mu-\operatorname{div}\phi(u) & \text{in }\Omega,\\\(4pt\) Bu+\lambda u=h & \text{on }\partial\Omega, \end{cases}{−Au+g(x,u,∇u)+δ∣u∣p0(x)−2u=μ−divϕ(u)Bu+λu=hin Ω,on ∂Ω,

where the right-hand side μ\muμ belongs to L1(Ω)+W−1,p⃗′(x)(Ω‾)L^1(\Omega)+W^{-1,\vec{p}'(x)}(\overline{\Omega})L1(Ω)+W−1,p′(x)(Ω), the operator AuAuAu is a Leray–Lions anisotropic operator and ϕ∈C0(R,RN)\phi\in C^0(\mathbb{R},\mathbb{R}^N)ϕ∈C0(R,RN), the nonlinear term g:Ω×R×RN⟶Rg:\Omega\times\mathbb{R}\times\mathbb{R}^N\longrightarrow\mathbb{R}g:Ω×R×RN⟶R satisfying some growth condition but no sign condition. We provide an existence result of entropy solutions for this class of anisotropic problems.

Mots-clés

entropy solutions, anisotropic elliptic problems, variable exponent, Fourier boundary conditions, measurement data, Leray-Lions operators, existence of solutions, nonlinear elliptic equations