Existence Of Solutions For Choquard Equations With Hardy-Littlewood-Sobolev Lower Critical Exponent

In this paper,a Brezis-Nirenberg problem of Choquard equations with lower critical exponent and the existence of groundstates to Choquard equations with weighted potentials and Hardy-Littlewood-Sobolev lower critical exponent are studied respectively by using variational methods.This thesis is divided into five chapters.It is organized as follows:In Chapter 1,we give a brief introduction to the background and research status of our problems.Meanwhile,some related preliminary knowledge and lemmas are given.The main results obtained in this thesis are also presented.In Chapter 2,we consider the following Brezis-Nirenberg problem of Choquard equations with lower critical exponent-?u=?u+(I?*|u|2?,*)|u|2?,*-2u,x??,where ? is a bounded domain with Lipschitz boundary in RN.Let ? ?(0,?1),N?3.Ia is the Riesz potential given for each x ? RN\{0},??(0,N).Thanks to variational methods,the mountain pass lemma,boundedness of Palais-Smale sequences and compactness,the existence of groundstates is obtained.In Chapter 3,we aim to study the existence of groundstates for Choquard type equations with weighted potentials and Hardy-Littlewood-Sobolev lower critical exponent in the following form-?u+V(x)u=(I?*[Q(x)|u|N+?/N])Q(x)|u|?/N-1u,x?RN,where V,Q?C(RN,R),N?3.Firstly,the existence of positive groundstates to the problem is obtained by using variational methods and assuming necessary conditions on V and Q.Then,we show that there is a non-existence result of nontrivial solutions for the problem by using the Hardy-Littlewood-Sobolev inequality and a Pohozaev identity.In Chapter 4,We investigate the existence of groundstates to the following form in an indefinite case-?u+V?,v(x)u=(I?*[Q?(x)|u|N+?/N])Q?(x)|u|?/N-1u,x?RN,we obtain the existence of groundstates to the problem for ?>?v by using a linking critical point theorem.In Chapter 5,we summarize the results of this thesis and make some discussions.