Existence and multiplicity of solutions for some Styklov problem involving p(x)-Laplacian operator. Issue 7 (3rd May 2022) - Chammem, R.; Ghanmi, A.; Sahbani, A.
Abstract : In this paper, we consider a class of p ( x ) -Laplacian problems of the form: 1 ( − Δ ) p ( x ) u + a ( x ) | u | p ( x ) − 2 u = f ( x, u ) in    Ω, | ∇ u | p ( x ) − 2 ∂ u ∂ v + b ( x ) | u | q ( x ) − 2 u = g ( x, u ) on    ∂ Ω, where Ω ⊂ R N, N ≥ 2 is a bounded domain with Lipschitz boundary ∂ Ω, ( ∂ / ∂ v ) is outer unit normal derivative. The functions a, b, p, q, g and f are assumed to satisfy suitable assymptions. The existence and the multiplicity of solutions is obtained by using variational methods, and mountain pass lemma combined with Ekeland variational principle.