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 ?>0 and q??2,6?,such that for all s>0,f?s?^?s^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.