Existence Of Positive Solutions For Non-local Problems In R~N

The existence of positive solutions for non-local problems are investigated by using varia-tional methods.Firstly,the existence of positive solutions for a class of Kirchhoff-Shrodinger-Poisson systems is investigated.Secondly,existence of positive solutions for a class of p-kirchhoff equations is investigated.The main theoretical bases are Mountain pass lemma and the method of combining Pohozaev manifold with Nehari manifold.In the second chapter,we consider the following Kirchhoff-Shrodinger-Poisson systemwhere a>0,b? 0 are constants,q>0 is a paremeter,1<p<5,K,f:R3?R satisfy the following conditions:The main results of the chapter are as follows.Theorem 2.1.1.If(f1)-(f2)and(K1)-(K2)hold,then there exists q0>0 such that,for any 0<q<q0 the problem(SK)possesses a positive solution(u,?u)? H1(R3)×D1,2(R3)for all 1<p<5.Theorem 2.1.2.If(f1)-(f2)and(K2)-(K4)hold,then the problem(SK)' possesses a positive ground state solution(u,?u)? H1(R3)×D1,2(R3)for all 1<p<5.In the third chapter,we investigate the following p-Kirchhoff type equation where a>0,b? 0 are constants,?pu=div(|?u|p-2?u)is the p-Laplacean operator,N? 3,1<p<N,f satisfies the following conditions:The main result of the chapter is as follows.Theorem 3.1.1.If(f1)'-(f5)' and(?1)-(?2)hold,then the problem(P)possesses a positive solution u ? Wr1,p(RN)satisfying I(u)>0.The structure of this paper is as follows.In the chapter 1,the essential ideology of variational methods and the recent development of Kirchhoff type problems by variational methods are introduced.Moreover,the research work and main results of this paper are stated.In the chapter 2,the preliminaries to prove the existence of positive solutions for Kirchhoff-Shrodinger-Poisson system in R3 are presented.More importantly,the proofs of the main results are given.In the chapter 3,the preliminaries to prove the existence of positive ground state solution for p-Kirchhoff type problem in RN is presented.More importantly,the proof of the main result is given.