Existence And Asymptotic Behavior Of Least Energy Positive Solutions To Local And Nonlocal Schr?dinger System

In the past two decades,Schrodinger system has been widely used in many physical problems such as Bose Einstein condensation and nonlinear optics,thus many mathematicians are interested in it.In this dissertation,we mainly study the existence and asymptotic behavior of least energy positive solutions of local and nonlocal Schrodinger systems by using the variational methods and elliptic equation theory.In Chapter 1,we introduce the background and the main results of this thesis.In Chapter 2,we introduce some of the symbols and knowledge used in this thesis.In chapter 3,we study the following nonlocal Schrodinger equations with Choquard type nonlinearities:where ?(?)RN is a smooth bounded domain,-?1(?)<?1,?2<0,?1(?)is the first eigenvalue of(-?,H01(?)),?1,?2>0 and ??0 is a coupling constant.We show that the critical nonlocal elliptic system has a positive least energy solution under appropriate conditions on parameters via variational methods.Moreover,the asymptotic behaviors of the positive least energy solutions as ??0 are studied.In Chapter 4,we consider the coupled Choquard system with general critical exponents:where ?(?)RN is a smooth bounded domain,2?*:=2N-?/N-2 is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality,-?1(?)<?1,?2<0,?1(?)is the first eigenvalue of(-?,H01(?)),?1,?2>0 and/??0 is a coupling constant.We show that the critical nonlocal system has a positive ground state solution when? is negative or ? is larger than some positive constant via variational methods.Furthermore,we study the phase separation phenomena of the limit profile of the positive ground state solution as ??-?,it seems to be first results for Choquard system in critical case.Moreover,some different phenomenon arises comparing with the local Schrodinger system.In Chapter 5,we investigate the existence of solutions to the following Schrodinger system in the critical case(?)in ?,ui=0 on(?)?,i=1,...,d,where,?(?)R4 is a smooth bounded domain,d?2,-?1(?)<?i<0 and ?ii>0 for every i,?ij=?ji for i?j,where ?1(?)is the first eigenvalue of-? with Dirichlet boundary conditions.Under the assumption that the components are divided into m groups,and that ?ij>0(cooperation)whenever components i and j belong to the same group,while<0 or is positive and small(competition or weak cooperation)for components i and j belonging to different groups,we establish the existence of nonnegative solutions with m nontrivial components,as well as classifi-cation results.Moreover,under additional assumptions on we establish existence of least energy positive solutions in the case of mixed cooperation and competition.The proof is done by induction on the number of groups,and requires new estimates comparing energy levels of the system with those of appropriate sub-systems.In the case ?=R4 and ?1=...?d=0,we present new nonexistence results.This paper extends and improves some results from[Chen-Zou,Arch.Ration.Mech.Anal.205(2012),515-551].