Existence and uniqueness of weak and entropy solutions for homogeneous Neumann boundary-value problems involving variable exponents

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Résumé

We study the nonlinear homogeneous Neumann boundary-value problem
b(u)−diva(x,∇u)=fin Ωa(x,∇u).η=0on ∂Ω,
where Ω is a smooth bounded open domain in ℝN, N≥3 and η the outer unit normal vector on ∂Ω. We prove the existence and uniqueness of a weak solution for f∈L∞(Ω) and the existence and uniqueness of an entropy solution for L1-data f. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

Mots-clés

elliptic equation; weak solution; entropy solution; Leray-lions operator; variable exponent