Existence Of Solution For The Critical Hartree System

In this thesis,the author applies the variational methods to study the existence of solution for the critical problem for Hartree system.In Chapter 1,the author introduce the research background and recent development of the critical Hartree system.Then we recall some preliminary knowledge and state the main results obtained in the thesis.In Chapter 2,first the author establish a result for the critical Hartree equation under the perturbation of both sublinear and suplinear terms.where ? is a smooth bounded domain of RN,0<s<1,N?3;0<q<1;1<p<2*s-1,?>0.We get the existence and multiplicity of solution for the problem.Then,the author investigate the existence of the positive solution for the critical Hartree equation where-?1(?)<?<0 with ?1(?)the first eigenvalue of(-?)s under the Dirichlet boundary condition.In Chapter 3,first the author are interested in the following critical coupled Hartree system where 0<s<1,?1,?2>0,??0,4s<?<N?N?3,2?,s*=(2N-?)/(N-2s).Assume that the nonlinearity and the coupling terms are both of the upper critical growth due to the Hardy-Littlewood-Sobolev inequality,we are able to obtain the existence of ground state solution of the critical coupled Hartree system.Then,the author are also going to consider the following critical coupled Hartree system where ? is a smooth bounded domain of RN,-?1(?)<?1,?2<0 with ?1(?)the first eigenvalue of(-?)s under the Dirichlet boundary condition.We can prove the existence of ground state solution of the critical coupled Hartree system.In Chapter 4,the author give some problems for further exploration.