Existence And Classification Of Ground State Solutions For Several Classes Of Nonlocally Coupled System

The existence and classification of ground state solutions for nonlocal equation-s(systems)are the important module in the filed of partial differential equations.In 2007,Caffarelli and Silvestre[1]given an extension method,which can transform the nonlocal problem into local problem,after that,there are lots of studies about the existence of ground state solutions for fractional Laplacian equations(systems).The main methods to study the existence of solution for nonlocal equations are variation method,finite dimensional and infinite dimensional reduction methods.This dissertation mainly through variational method,mountain pass lemma,Ekeland's variational principle,implicit function theorem,Vatali's theorem and the method of convert-ing to an algebra system to study the existence of ground state solutions for critical nonlocal coupled system,the existence of ground state solutions with one critical exponent and one subcritical exponent,the existence of ground state solutions for fractional Laplacian system with sign-changing weight functions,the existence and classification of ground state solu-tions with different Moser index for subcritical system.The full text is divided into seven chapters,the first two chapters introduce the research background,research states and some preliminaries,from chapter four to chapter six,we give the details of the proof for four aspects mentioned above.First of all,we give the existence of ground state solutions for critical coupled system in full space.However,if(?)is a solution of system,then(k,l)must satisfies an algebra system.Based on above observation,we transform the problem of existence of ground state solutions for critical system to the existence of solutions for an algebra system,then by studying the algebra system to get the existence of ground state solutions under different conditions.When we consider the critical system on bounded domain of RN,we first show that the system has mountain pass structure,by the mountain pass lemma we obtain the existence of Palais-Smale sequence.In order to prove the convergence of Palais-Smale sequence,we first prove that the Palais-Smale sequence are bounded,by the Sobolev embedding theorem,we can obtain the weakly convergence of the Palais-Smale sequence,finally,by Brezis-Lieb lemma and energy comparison,we obtain the strongly convergence of the Palais-Smale sequence.Afterwards,we consider nonlocal coupled system with one critical exponent and one subcritical exponent in full space.Since for all p?1,HS(RN)?LP(RN)are not compact embedding,in order to overcome the lack of compactness,we deal with in sym-metric space,then by the principle of symmetric criticality,we get the existence of ground state solution for original space.On symmetric space,by the method of variational method,mountain pass lemma,Sobolev embedding theorem,Vatali's theorem and energy compari-son method to get the existence of ground state solutions for coupled nonlocal system with one critical exponent and one subcritical exponent,that is we prove there exist a ?0?(0,1),such that when 0<???0,system has a positive ground state solutions.When ?>?0,there exist a ??,(?),such that if ?>?u,v,system has a positive ground state solution,if ?<?u,v,system has no ground state solution.In the next part,we consider the coupled system with sign-changing weight functions,by the method of Nehari manifold decomposition,Ekeland's variational principle,implicit function theorem and Taylor expansion,we prove the existence of Palais-Smale sequence,then we prove the convergence of the Palais-Smale sequence.When the pair of(?,?)be-longs to the certain subset of R2,we prove that system has at least two positive solutions.In the end,we consider the existence and classification of ground state solution for subcritical coupled system.Under suitable conditions of ?,?,?,?1,?2,we give a complete classification of ground state solutions with different Moser index and prove that if(u0,v0)is any positive ground state solution,then we have(?)with different Moser index under suitable conditions.Finally,we give the main steps of the proof for finite and infinite dimensional reduction and give some questions may be study by the methods of finite and infinite dimensional reduction.