Existence And Multiplicity Of Solutions For Fractional Kirchhoff Equations With Critical Nonlinearity

Recently,nonlocal partial differential equation has a very wide application in many areas,such as continuous media mechanics,the phase image,population dynamics and the game theory,etc.In the field of nonlocal equation in theoretical physics,financial system and fluid mechanics has been widely used.The fractional order differential equation is a generalization of the integer order equations.It can more accurate to depict the natural phenomenon,and to the simulation of the dynamic process of life better than integer order partial differential equat.ions.In this paper we consider some kinds of Kirchhoff equations with critical nonlinearity.We obtain the existence of non-trivial solution and multiple solutions by using the variational methods under suitable conditions.This paper is organized as follows.In the first chapter,we give the background and significance of nonlocal partial differential equation,and introduce the structure of the dissertation and some prelim-inaries.In chapter 2,we study the following fractional Schrodinger equations with critical nonlinearity of the form where potential function V(x)has critical frequency,that is rninx?1RNV(x)? 0.The lack of compactness conditions is caused by the critical nonlinear term.We will use the concentration compactness principle to overcome the difficulties,the method in this paper is very different with previous literatures[42,137].More importantly,our method in this paper is also suitable for other potential functions.In the end,we obtain the existence of nontrivial solutions and multiple solutions by using the variational method.Chapter 3 is to consider the following critical Kirchhoff type equations involving the fractional p-Laplacian operator where ?(?)RN is an open bounded domain with smooth boundary.On the one hand,the lack of compactness conditions is caused by the critical nonlinear term.We will use the fractional version of concentration compactness principle to overcome the difficulties.On the other hand,because of the energy functional neither upper nor lower bound,we will use the truncation method to solve this difficulty.We prove the existence of infinitely many solutions of this problem which tend to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya[66].In the chapter 4,we consider the fractional Schrodinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity in RNWe can prove that compactness conditions by choosing appropriate parameters ? and?,together with the fractional version of concentration compactness principle.We can also prove the existence of infinitely many solutions of this problem which tend to zero by using a new version of the symmetric mountain-pass lemma due to Kajikiya[66].