Existence Of Multiple Solutions For A Class Of Semilinear Elliptic System

In recent years,the research of semilinear elliptic equations has attracted more and more people's attention,on the one hand,this is because this kind of problem often comes from many important research of nonlinear phenomena,such as population problem,chemical reaction,optical research,etc.;on the other hand is because the study of equations is the continuation of a single equation.In this paper we study the elliptic system as follows form:where?is a smooth bounded domain in RN(N?3);?>1,?>1 satisfying?+??(2,2*),where 2*denotes the critical Sobolev exponent,that is 2*=2N/N-2;Q?L?(?),and Q(x)?0 a.e.in?;hi(x)?L2(?),(i=1,2)are nonzero functions.We use the variational method to prove the conclusion as follows:Let?be a smooth bounded domain in RN(N?3);?>1,?>1 satisfying?+??(2,2*),where 2*denotes the critical Sobolev exponent,that is 2*= 2N/N-2;Q?L?(?),and Q(x)?0 a.e.in?.Then there exists?*>0 such that for every nonzero function hi(x)?L2(?),(i=1,2),with(??(h12+h22)dx)1/2??*,problem(*)admits at least two nontrivial solutions.