Solutions Of Nonlinear Elliptic Systems With Coupled Items

In recent years,partial differential equations and systems which originated from physical problems have received much attention from science researchers.Many math-ematicians and physicians have deeply investigated in nonlinear elliptic equations and systems and plenty of excellent results have come out successively,for example,exis-tence,uniqueness and multiplicity of solutions to the systems.Heretofore there are few results concerning Bose-Einstein condensate related k--coupled systems due to the complicated structure.Existence,uniqueness,multiplicity,asymptotic behaviour and other qualitative results of nontrivial solutions to the k--coupled systems remain to be investigated.Meanwhile,much difficulties arises from the existence of parameters related algebraic systems when classifying the solutions.In this thesis,we aim to study the existenceand qualitative results of nontrivial solutions,as well as the the least energy estimates of the Bose-Einstein condensate related systems,via variational methods and classical elliptic equations theories.Firstly,we consider the critical k--coupled nonlinear Schr?dinger systems in dimen-sion N?5.We introduce the idea of induction to get the refined estimates of the least energy and obtain the existence results of positive ground state solutions to the critical k--coupled nonlinear Schr?dinger systems and related limit systems if all coupling constants are positive.Meanwhile,we give the first answer to the existence of positive solutions to the algebraic system and subsequently illustrate the ground state solutions of synchronized type,where an ingenious idea is used to transform the existence of positive solutions of the algebraic systems to the existence of infimum of a geometric problem.These results generalize those of Chen,Luo,Wu and Zou,and give the first classification results of pos-itive ground state solutions of the the critical k--coupled nonlinear Schr?dinger systems in dimension N?5.Besides,we generalize these results to the the critical k--coupled nonlinear Schr(?)dinger systems involving fractional Laplace in dimension N> 4s.We system-atically investigate in the existence and non-existence of ground state solutions.Since the nonlinear systems involving fractional Laplace are more complicated,we shall introduce some new techniques.Meanwhile,we are interested in the asymptotic behaviour of the solutions.We generalize the results of Chen and Zou to the systems involving fractional Laplace and obtain the existence results of another system at the same time.Finally,we consider the normalized solutions to the nonlinear coupled Schr?dinger systems.There are few results involving systems.The existence of normalized solutions can be obtained by studying the critical points of energy functional under some constraints.The systems can be treated as a linear perturbation of BEC systems,which arises much difficulties when dealing with the compactness of minimizing sequences.We obtain the compactness of minimizing sequences via Shibata's rearrangement and Schwarz's rear-rangement,and further we prove the existence of normalized solutions.