Existence of solutions for a class of nonlinear type problems involving the p(x)-Laplacian operator

Abstract

The aim of this paper is to establish the existence of at least one nontrivial weak solution of the following problem −∆ p(x) u + α(x)|u| p(x)−2 u = λ a(x)|u| q(x)−2 u in Ω, |∇u| p(x)−2 ∂ u ∂ ν + β (x)|u| p(x)−2 u = λ b(x)|u| r(x)−2 u on ∂ Ω. Our technical approach is based on the direct variational method combined with the mountain pass theorem and the theory of Lebesgue and Sobolev spaces.