Existence And Multiplicity Of Solutions For Several Choquard Type Equations With Critical Exponent

In this paper,existence and multiplicity of solutions for several Choquard type equations with critical exponent are studied via variational methods and critical point theory.The main content of this article is as follows:Chapter 1 mainly introduces the research background and significance of the Choquard type equation and the Kirchhoff type equation,as well as the research present situation.In chapter 2,we give some basic function spaces and properties,and some abstract critical point theorems.In chapter 3,we study the existence of ground state solutions for a class of linearly coupled systems of Choquard type with lower critical exponent.The non-negative ground state solution of the system is obtained by the method of Nehari manifold under the condition that the potential function satisfies a certain constraint,and the change tendency of the ground state solution when the parameter ? 0 is studied.In chapter 4,the existence of ground state solutions for a class of nonlinearly coupled systems of Choquard type with lower critical exponent is studied.Firstly,the existence of the ground state solution for the limit problem is obtained by estimating the energy level of the mountain pass corresponding to it.Furthermore,the existence of the ground state solution of the original problem is obtained by comparing the functional energy of the original problem with that of the limit problem.In chapter 5,we study the multiplicity of solutions for a class of Kirchhoff type equations with Hardy–Littlewood–Sobolev critical nonlinearity.The difficulty of overcoming the lack of compactness in critical problem is mainly based on the second concentrated compactness principle,and the multiplicity of nontrivial solutions of the original problem is obtained by using a deformation of the symmetric mountain pass theorem and truncation technique.