| This paper studies the existence of nontrivial positive solutions for a class of fractional Schrodinger equations with weight function(?)where 0<s<1,N>2s,λ∈R,1<p<2(s)*-1,2(s)*=N2/N-2s.The main work of this paper is to prove the existence of nontrivial positive solutions of the equations.Therefore,two conditions are proposed for the weight functions ki(x),i=1,2.Specifically,the equations admit a positive ground state solution when λ∈(0,1)and a higher positive energy solution when λ is very small as long as the weight functions satisfy one of the conditions.The main contents of this paper are as follows:In the first chapter,firstly it introduces the background and research status of the frac-tional Schr(?)dinger system,then introduces the main problems,conclusions and idea of this paper.In the second chapter,it introduces the basic knowledge which is necessary for the problem,including the knowledge of fractional Sobolev space,fractional Laplacian and variational methods.In the third chapter,as the basis for studying equations,we use the classical mountain pass lemma,Nehari manifold and the strong maximum principle to show the existence of the positive ground state solution of the single equation(-△)su+u=k(x)|u|p-1u,x∈RN,u∈Hs(RN).In the fourth chapter,we firstly prove that the equations have a positive ground state solution using the conclusion of chapter 3 and then construct a special(PS)sequence us-ing quantitative deformation lemma.Finally,we show the convergence point of the(PS)sequence is a positive higher energy solution of the equations.In the fifth chapter,we make a summary of the research contents and then put forward some prospects for the research problem,including the condition of the weight functions,the critical exponent cases and the regularity of the solutions. |
