| Choquard type equation appears in many physical models.It was first used to describe the polarization phenomenon of particles,and then it was widely used in quantum mechanics,boson star theory and other fields.Choquard equation is a powerful tool to reveal physical phenomena.At the same time,as a kind of partial differential equation,the existence of solutions and other properties of Choquard equations have become the focus of research.In this paper,we mainly study the existence of solutions for Choquard equations with critical growth under the disturbance of the variable potential and local nonlinear perturbation.By using the variational method and critical point principle such as mountain pass lemma,we construct a suitable workspace,and prove the existence of nontrivial solutions and ground-state solutions for several kinds of Choquard equations.Chapter 1 mainly introduces the research background and current situation of Choquard equation,and the Chapter 2 gives some preliminary knowledge such as lemmas and inequalities.In Chapter 3,we study the existence of ground state solutions for a class of Choquard equations with doubly critical exponents and local perturbation The Pohozaev identity is mainly used to overcome the lack of compactness caused by the doubly critical nonlinear term.We verified that the energy functional of this equation is unbounded from below and its unique critical point corresponds to its maximum point With the help of the mountain pass lemma,we proved that this kind of Choquard equations has at least a ground state solution.In Chapter 4,we investigate the existence of ground state solutions for a class of Choquard equations with low critical growth under the variable potential and local perturbation We mainly use the variational method to transform the related problems of equation solutions into the solution of functional critical points.First,we find an appropriate bounded domainM,and discuss the relationship between the limit of convergence sequence and the critical point of functional in the space of continuous function Secondly,by using some analytical techniques,an estimate of the energy functional m corresponding to the critical point was obtained.Finally,it is proved that the equation has a ground state solution and the range of constant values λ for the perturbation term in different cases is givenIn Chapter 5,we study the existence of semiclassical solutions for a class of Choquard equation with variable potential and Critical exponent Firstly,a suitable workspace is constructed by using Pohozaev equation to make the functional L(u)well defined.Secondly,we prove that this Choquard equation admits a ground state solution of mountain pass type without super-linear conditions near infinity on the nonlinearity.Then the minimum energy solution of the equation is discussed under the weaker condition of the potential variation.On this basis,it is proved that there is at least a minimum energy solution for the Choquard equation of semiclassical state. |
