The Study Of Solutions Of Elliptic Equations With Non-local Operators

In recent years,a great attention has been focused on the study of fractional laplace equation,especially its non-linear equation.In fact,fractional laplace oper-ator has special applications in many fields,such as,optimization,finance,stratified materials,anomalous diffusion and so on.And obtained many new results.In this paper,the existence of positive solutions,negative solutions,sign-changing solutions and infinitely many sign-changing solutions of elliptic equation with non-local operators is obtained by using the variational methods,the mountain pass theorem and a version of the symmetric mountain pass theorem.The dissertation contains four chapters.In chapter 1,we mainly introduce the research status of fractional laplace oper-ator equation and some basic knowledge and symbols commonly used in this paper.In chapter 2,we consider the following non-local elliptic euqation:(?),where (?) is the fractional laplace operator,defined as(?).Using a new version of the mountain pass theorem,we can get the existence of non-trivial solutions under suitable conditions.In chapter 3,we consider the positive,negative and sign-changing solutions of above non-local elliptic equation.In order to prove this problem,we use the vari-ational method and relies on the application of three critical point theorems.The first one is a version of the mountain pass theorem.More precisely,using the quanti-tave deformation lemma we derive a variant of the mountain pass theorem on cones.The second critical point theorem is a sign-changing version of the mountain pass theorem to ensure the existence of a sign-changing solution.The third critical point theorem is a version of the symmetric mountain pass theorem,which guaratees the existence of infinitely many high energy sign-changing solutions.In chapter 4,we consider the above non-local elliptic equation.By using the sign-changing critical point theorem,under some suitable conditions,we can obtain infinitely many sign-changing solutions.