Existence and multiplicity of solutions for p(x)-Kirchhoff-type problem in Rn

In this paper we study the p(x)-Kirchhoff-type problem { -m ( ? R^N 1/p(x) | ?u | ^ p (x) dx ) div ( | ?u | ^ p(x)-2 ?u ) + | u | ^ p(x) - 2 u = f(x.u) in R^N {u ? W^1, p(x) (R^N). We first establish the compact imbedding imbeddingW^1, p(x)(RN) ? Lq(x) b(x)(RN), where Lp(x) b(x) (RN) ={u is measurable on RN: ? RN b(x) |u| p(x)dx < ?}. Based on it, the existence and multiplicity of solutions for the problem are obtained by variational methods.