Solutions Of Nonlinear Schr(o|¨)dinger Systems

In the last decades, nonlinear Schro|¨dinger systems have received a lot of attention,since they have great applications to many physical problems such as nonlinear optics andBose-Einstein condensation. Many famous mathematicians have made a lot of excellentresearches on nonlinear Schro|¨dinger systems.In this thesis, we study the existence and qualitative properties of nontrivial solutionsfor some nonlinear Schro|¨dinger systems via variational methods and elliptic equationstheories.Firstly, we consider a Bose-Einstein condensation system with cubic nonlinearities.In the subcritical case (i.e. the spatial dimension is2or3), we study the optimal pa-rameter range for the existence of ground state solutions, the uniqueness and asymptoticbehaviors of ground state solutions. These give the first partial answer to an open ques-tion raised by Sirakov in (Comm. Math. Phys.,2007,271:199-221). We also obtainthe existence and related properties of infinitely many sign-changing solutions for anynegative coupling constants. In the critical case (i.e. the spatial dimension is4), we makea systematical research on ground state solutions of this problem, including the existence,the nonexistence, the uniqueness and the phase separation phenomena of the limit profileas the coupling constant tending to minus infinity. These seems to be first results for thissystem in the dimension4case.Besides, we can extend some results above to a general critical system (i.e. a homol-ogous critical problem with spatial dimensions≥5). It turns out that some quite diferentphenomenon happen comparing with the dimension4case. For example, we can provethe existence of ground state solutions for any nonzero coupling constant, which can nothold in the dimension4case. When the spatial dimensions≥6and the coupling constanttends to minus infinity, we prove that the ground state solutions of this system convergeto sign-changing solutions of the Brezis-Nirenberg critical exponent problem. Hence thisgeneral critical system is related closely to the well-known Brezis-Nirenberg critical ex-ponent problem. Here we study the uniqueness and sharp energy estimates of least energysolutions for the Brezis-Nirenberg critical exponent problem in a ball, which is very use-ful in the study of the above general critical system. We also obtain multiple solutions for the Brezis-Nirenberg critical exponent problem in a general smooth bounded domain.Finally, we consider Ambrosetti type linearly coupled Schro|¨dinger equations withcritical exponent. We study the existence and nonexistence of ground state solutions fordiferent coupling constants. Remark that our result is almost optimal.