Existence Of Solutions For Two Classes Of Schr(?)dinger-Kirchhoff-Poisson Systems

In this paper,the existence of solutions for two classes of Schr（?）dinger-Kirchhoff-Poisson systems is investigated by using variational methods.Firstly,the existence of nontrivial solutions for Schr（?）dinger-Kirchhoff-Poisson systems with a parameter is studied.Secondly,the existence of positive ground state solutions for Schr（?）dinger-Kirchhoff-Poisson systems involving critically growing nonlocal term is considered.The main theoretical basis are mountain pass theorem,general minimax principle,Br′ezis-Lieb lemma,monotonicity technique and some analytical skills.In chapter 2,the following Schr（?）dinger-Kirchhoff-Poisson system with a parameteris investigated,where(a>0,b≥0 are constants,m>0 is a parameter,2<p<6.The main result of the chapter is as follows:Theorem 1.2.1.The system（p₁）possesses a nontrivial solution for any m>0with p∈（4,6）,or a sufficiently large m>0 with∈(2,4].In chapter 3,the following Schr（?）dinger-Kirchhoff-Poisson systems involving criti-cally nonlocal termis considered,where(a>0,b≥0 are constants,and(V:R³→R is a potential function.（V,f）satisfy the following assumptions:（V₁）there exists a positive constant V_∞such that for all x∈R³,（V₂）(V∈C（R³,R）is weakly differentiable,satisfying the following inequality:where（·,·）is the usual inner product in R³,is the best Sobolev constant;（f₁）f∈（R,R）is odd,（f₂）（f₃）there exist constants μ>0 and q∈（2,6）,such that for all s>0,f（s）^μs^q-1.When V（X）≡1,the problem（P₂）can be reduced toThe main results of the chapter are as follows:Theorem 1.2.2.Let（f₁）-（f₃）hold.Then the system（f₃）possesses a positive ground state solution for any>0 with∈（4,6）,or a sufficiently large>0 with∈(2,4].Theorem 1.2.3.Let（1 and1）satisfy the conditions（V₁）-（V₂）and（f₁）-（f₃）respec-tively.Then the system（₂）possesses a positive ground state solution for any>0with∈（4,6）,or a sufficiently large>0 with∈(2,4].The structure of this paper is as follows:In chapter 1,the essential ideology of variational methods and the recent develop-ment of research development of Schr（?）dinger-Kirchhoff-Poisson systems by variational methods are introduced.Moreover,the research work and main results of this paper are stated.In chapter 2,some necessary preliminaries to prove the existence of nontrivial solutions for Schr（?）dinger-Kirchhoff-Poisson systems with a parameter are presented.More importantly,the proof of the main result is given.In chapter 3,some necessary preliminaries to prove existence of positive ground state solutions for Schr（?）dinger-Kirchhoff-Poisson systems involving critically nonlocal term are presented.More importantly,the proofs of the main results are given.