Existence and multiplicity of solutions for Dirichlet problem of p(x)-Laplacian type without the Ambrosetti-Rabinowitz condition

Abstract

This paper deals with the following variational problem with variable exponents{−div((1+|∇u|p(x)1+|∇u|2p(x))|∇u|p(x)−2∇u)=f(x,u),inΩ,u=0,on∂Ω, where Ω⊆RN (N≥2) be a bounded domain with the smooth boundary ∂Ω, p:Ω‾→R is a Lipschitz continuous function with 1<p−:=essinfx∈Ω⁡p(x)≤p(x)≤p+:=esssupx∈Ω⁡p(x)<N and f∈C(Ω×R,R) is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz type condition. Under three different superlinear conditions on f at infinity, we prove that the above equation has at least a nontrivial solution. Moreover, the existence of infinite many solutions is proved for odd nonlinearity. Especially, some new tricks are introduced to show the boundedness of Cerami sequences.