Solvability Of Elliptic Equations With P(x)-Laplacian-like Operator

In this paper,the existence and multiplicity of solutions to problems with p(x)Laplacian-like operator are studied by using the variational methods such as symmetric mountain pass lemma,space decomposition technique and Fountain Theorem.Firstly,we consider the following elliptic equations with p(x)-Laplacian-like operator (?) where N ≥ 2,Lu=div(1+(?))|▽u|p(x)-2▽u),V:RN→(0,∞)is a continuous function,p:RN→(1,∞)with 1<p-:=(?)p(x)≤p+:=(?)p(x)<N.Under the condition that the nonlinear term f(x,u)does not satisfy the(AR)condition,by proving that the functional of such problems satisfies Cerami condition,the existence of infinitely many solutions for such problems are obtained by using symmetric mountain pass lemma and variational method.Secondly,we consider the following nonlocal Neumann problem with p(x)-Laplacian-like operator (?) where Ω(?)RN is a bounded domain with smooth boundary (?)Ω,a(t)∈C(R-,R+),f(x,u)∈ C(Ω×R,R),(?)v is the outward unit normal on (?)Ω,p∈ C+(Ω)={h|h∈C(Ω)h(x)>1,(?) x∈Ω} with 1<p-:=(?)p(x)≤p(x)≤p+:=(?)p(x)<N.The existence and multiplicity of solutions for this kind of problems are obtained by using the critical point theory such as space decomposition technique and Fountain Theorem.