Research On The Existence Of Multiple Solutions Of Elliptic System With Perturbations

In recent years,the existence of solutions to nonlinear elliptic equations by using variational methods have drawn many mathematicians' attention.For practical problems,the perturbation is always inevitable.Therefore,it is meaningful to study the problem of elliptic equations with perturbation.In this thesis,we mainly study the existence of multiple solutions for two classes of elliptic equations with perturbations.These two kinds of equations are derived from the physics and chemistry.It has important application background and theoretical value.Specifically,on one hand,we study the existence of multiple solutions for p-Laplacian equations with perturbations.On the other hand,we study the existence of multiple solutions of nonlinear coupled Hartree type equations with perturbations.In this thesis,the variational method is used to prove the main results.First,to guarantee the variational functional is bounded from below,we restrict it on a set N(usually called Nehari manifold).Second,in order to prove the existence of multiple solutions of the equations,the set is divided into three parts N~+,N~0 and N~- by using fiber map.Moreover,we prove the existence of minimum in N~+ and N~-by studying the properties of the two parts.Finally,we prove that the constrained minima are the critical point of whole space.If the perturbations h_i(x)(i=1,2) are positive,there exist a positive ground state solution and a positive bound state solution.Therefore,we prove the existence and properties of solutions of the equations.The results of this paper provide the basis for the further study of the existence of multiple nontrivial solutions of the critical growth of p-Laplacian equations and Hardy-Littlewood-Sobolev critical growth of nonlinear coupled Hartree type equations with perturbation.