Infinitely Many Solutions for a Class of Kirchhoff Problems Involving the p(x) -Laplacian Operator

This article is devoted to studying a class of generalized p(x) -Laplacian Kirchhoff equations in the following form: -M(∫_Ω1/p(x)|∇ u|^p(x))div(|∇ u|^p(x)-2∇ u)=λ |u|^r(x)-2u +f(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain of ℝ^N (N≥ 2) with smooth boundary ∂Ω , λ>0 , and p and r , are two continuous functions in Ω . Using variational methods combined with some properties of the generalized Sobolev spaces, under appropriate assumptions on f and M , we obtain a number of results on the existence of solutions. In addition, we show the existence of infinitely many solutions in the case when f satisfies the evenness condition.