Existence Of Solutions For P(x)-Laplacian Equations On R～N

The aim of this paper is to study the existence and multiplicity of solutions for p(x)-Laplace equation in R~N .The study of the existence of solution for p(x)-Laplactan equation is an important field in the nonlinear analysis with variational exponents and that because of the lack ofr compactness for the unbounded domain,the above problem in R~N became more com-plex.This paper is mainly based on the works of Fan Xianlin,aim at the rank of the nonlinear item,we divided the problem into three cases.the first case is that the rank of the nonlinear item is lower than p(x),the second is that which is higher than p(x),and the third case is the combination of the above two cases,in this paper ,we respectively get the existence of solutions for the p(x)-Laplacian equation in these three cases.and get the existences and multiplicities of the positive solutions,negative solutions ,radial solutions and nonradial solutions for the relevant p(x)-Laplacian equations in R~N by the means of principle of symmetric criticality ,genus theory ,mountain pass theorem,mountain pass theorem in the symmetric case,fountain theorem,dual fountain theorem and so on .