On a p(x)-Kirchhoff equation via variational methods

This paper is concerned with the existence of two nontrivial weak solutions for a p(x)-Kirchhoff type problem of the following form ( −M R Ω 1 p(x) |∇u| p (x) dx ∆p(x)u = λ(x)|u| q(x)−2u in Ω, u = 0 on ∂Ω, by using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland’s variational principle and the theory of the variable exponent Sobolev spaces.