The Existence Of Solutions For Several Kinds Of Elliptic Equations Based On Variational Method

The elliptic equation plays a more and more important role in the development of natural science,especially in the fields of physics such as fluid mechanics,elasticity,electromagnetism and other fields of sciences.In this paper,several kinds of elliptic equations are studied by using variational method and critical point theory,and obtain a series of results about the existence and uniqueness of sign-changing solutions and infinitely many high-energy solutions,this paper extends and improves the current literature on the existence of relevant conclusions.The main results are summarized as follows:In Chapter 1,we introduce history and current research states of the variational method and many experts and scholars of the application results.At the same time,we give the structure of the paper,the relevant theoretical basis,as well as our common convention symbolsIn Chapter 2,we investigate the existence of sign-changing solution for a class nonlocal Kirchhoff type equation#12 where a and b are positive constants.With the help of the constraint variational method and via a direct approach,we show the existence of sign-changing solution and prove that the sign-changing solution has precisely two nodal domains.This work can be regarded as the complement for some results of the literature.In Chapter 3,we are concerned with a class nonlocal Kirchhoff equation with the term of Choquaxd#12 where a and b are positive constants.With the help of Hardy-Littlewood-Sobolev inequality,we show the existence of the bounded convergent(PS)c sequence.Combining with the Mountain pass theorem,we prove the existence of a nontrivial solution for the class nonlocal Kirchhoff equation with the term of Choquard.Furthermore,we also obtain at least one least-energy sign-changing solution via Hardy-Littlewood-Sobolev inequality and Brouwer topological degree.In Chapter 4,we study the existence of nontrivial solutions for a class of Kirchhoff type problems with negative coefficient.By means of Dirichlet principle and symmetrical Mountain-Pass Theorem,the existence of at least a nontrivial solution,a local negative energy nontrivial solution and a global positive energy nontrivial solution are obtained.In Chapter 5,we deal with a class of fractional Schrodinger-Poisson systems#12 where s ?(3/4,1),p ?(3,5),? is a positive parameter.By the variational method,we show that there exists ?(?)>0 such that for all ??[?1,?1,+?(?)),the above fractional Schrodinger-Poisson systems possesses a nonnegative bound state solution with postive energy.Here ?1 is the first eigenvalue of(-?)s+V(x).In Chapter 6,we consider a class of nonlinear fractional Schrodinger systems#12 where s ?(0,1),N>2.Under relaxed assumptions on V(x)and F(x,u,v),we show the existence of infinitely many high energy solutions to the above fractional Schrodinger systems by variant fountain theorem.