Inhomogeneous Elliptic Systems Involving Hardy-type Terms And Critical Sobolev Nonlinearities

In this paper,existence of multiple solutions to a system of inhomogeneous elliptic equations is investigated,which involves Hardy-type terms and critical Sobolev nonlinearities.This paper is divided into the following parts:In the first chapter,we introduce the relevant content and background of the research objectives,give the notations and the relevant preliminary results,introduce the main conclusions,and list the main structural arrangement.In the second chapter,first five related lemmas are given,by using Ekeland's variational principle and the implicit function theorem.By the lemmas,we can get there exsists a minimizing sequence ?(un,vn)?(?)?,satisfying the following conditions I(un,vn)?c0 and ?I'(un,vn)?(H2)-1=0 as n??,under the parameters satisfy certain constraints.Then the existence of(u0,v0)the first solution for the equations is proved,whenf(x),g(x)the inhomogeneous functions satisfy certain constraints.In the third chapter,the existence of the second solution for the equations is mainly studied.Considering the singularity of the equations solutions at the origin,we take to establish local Palais-Smale condition.Firstly,it is verified that(u0,v0),the solution obtained in the second chapter is a local minimizer of I(u,v).Secondly,by the concentration compactness principle,we can obtain that there exists a sequence,satisfying(un,vn)?(u,v)as n? ?.Finally,when sup/s?0I(su,Sv)<C0+1/NS(?1,?2)N/2 is established,the existence of(u,v)the second solution for the equations is verified,and(u,v)?(u0,v0).