Existence And Multiplicity Of Solutions For Elliptic Systems With Nonstandard Growth Condition In R～N

In this paper, we main study the existence and the multiplicity of solutions for elliptic systems with nonstandard growth condition in R N. This paper is based on the basic knowledge of the generalized Orlicz space and the generalized Orlicz-Sobolev space. With the studying of the form of Euler-Lagrange functional corresponding to the elliptic systems and it's properties of weak-strong continuity, ( S+ ), PS-compactness, Mountain-Pass geometry, etc., we managed to apply the Mountain-Pass Lemma in Critical Point Theory to prove the existence of solutions for elliptic systems with nonstandard growth condition in R N, under"sublinear"and"superlinear"conditions. By the Genus Theory and Fountain Theorem, we respectively prove the multiplicity of the solutions for elliptic systems with nonstandard growth condition in R N under"sublinear"and"superlinear"conditions.This paper contains five sections. In section one, we introduced the problem we are going to study and the present researching situation abroad. In section two, we present some necessary preliminaries about the generalized Orlicz space and the generalized Orlicz-Sobolev space. In section three, we first present the Euler-Lagrange functional corresponding to the elliptic systems that are studied in this paper, and its properties. Secondly, we prove the lemmas that will be used in proof of the following theorems and give the remark 3.1 that is very important in this paper. In section four, through the Mountain-Pass Theorem in Critical Point Theory, we respectively prove the existence of solution for elliptic systems with nonstandard growth condition in R N, under"sublinear"and"superlinear"conditions. In the last part of this paper, by the Genus Theory and Fountain Theorem, we respectively prove the multiplicity of the solution for elliptic systems with nonstandard growth condition in R N under"sublinear"and"superlinear"conditions.