On the superlinear problems involving the p(x)-Laplacian and a non-local term without AR-condition

In this paper, we consider a kind of nonlinear eigenvalue problem without Ambrosetti-Rabinowitz condition, that is, {−η[u]div(|∇u|p(x)−2∇u)=λf(x,u),a.e. in  Ω,u=0,on  ∂Ω, where Ω⊂RN is a bounded domain, p:Ω¯→(1,+∞) is a continuous function, η[u] is a non-local term defined by the following relation η[u]=2+(∫Ω1p(x)|∇u|p(x)dx)p+p−−1+(∫Ω1p(x)|∇u|p(x)dx)p−p+−1. Here λ>0 is a parameter, and f(x,u) is (p+)2p−-superlinear at infinity. Existence of nontrivial solution is established for arbitrary λ>0. We first prove the existence of nontrivial solutions of the system for almost every parameter λ>0 by using Mountain Pass Theorem, and then we consider the continuation of the solutions.