Existence Of Multiple Solutions For A Class Of Nonlinear Choquard Equations With Hardy-Littlewood-Sobolev Lower Critical Exponents

This dissertation focuses on the existence of multiple solutions to a class of nonlinear Choquard equations with Hardy-Littlewood-Sobolev lower critical exponentials.The full text is divided into five chapters,with specific contents arranged as follows:In Chapter 1,the research background and current situation of Choquard equation are described,and the symbols used in this paper are explained.In Chapter 2,we introduce the prerequisites and main theorems used in this paper,and present the main results.In Chapter 3,we mainly analyze the compactness of the following Palais-Smale sequences corresponding to the energy functional J_μ,v of the nonlinear Choquard equation with the lower critical index of Hardy-Littlewood-Sobolev,（?） WhereI_α is Riesz potential,α∈（0,N）,N≥3,μ∈R,v>0,V_μ,v=1-μ/（v²+|x|²）,J_μ,v（u）=1/2∫_R^N(|▽u|²+V_μ,v|u|²)dx-N/（2（N+α））∫_R^N（I_α*|u|（N+α）/N）|u|（N+α）/Ndx.By using BrezisNirenberg’s method of localized energy functional level set and combining with Lions’theorem of concentrated compactibility,the compactibility of Palais-Smale sequences is obtained.In Chapter 4,we mainly study a class of nonlinear Choquard equations with Hardy-Littlewood-Sobolev lower critical exponentials existence （?） Where I_α is Riesz potential,α∈（0,N）,N≥3,μ∈R,v>0,V_μ,v=1-μ/（v²+|x|²）.By using variational method,（▽）-theorem,surrounding the theorem,we get in the H¹（R^N）,there are at least three nontrivial solutions of the problem.In Chapter 5,we summarize the main contents of this study and prospect the further study of Choquard equation in the future.