Existence and multiplicity of solutions for some Styklov problem involving <i>p</i>(<i>x</i>)-Laplacian operator

Abstract

In this paper, we consider a class of p(x)-Laplacian problems of the form: (−Δ)p(x)u+a(x)|u|p(x)−2u=f(x,u)in Ω,|∇u|p(x)−2∂u∂v+b(x)|u|q(x)−2u=g(x,u)on ∂Ω, where Ω⊂RN, 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.

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