Solutions Of Bose-einstein Condensates Related Systems

During the last decades, nonlinear Schr ¨odinger systems have attracted much attention from a lot of science researchers because of the strong physical background. Among them, the Bose-Einstein condensates(BEC) system especially stands out. For scientific study, mathematicians and physicists are very interested in the problem about the existence of solitary wave solutions. Lots of excellent results have come out one by one.Among them, many problems related to mathematics are challenging and have important academic value.This thesis aims to study solutions of the BEC related systems by taking use of variational methods and elliptic equations theories, including the least energy estimates and the problem about the existence of ground state solutions, infinitely many solutions and sign-changing solutions, as well as the non-existence of solutions.Firstly, we consider the dimension N ≤ 3 subcritical BEC system. We construct an abstract theorem which implies that if a functional has a critical point in some set, then after “a small perturbation”, the new functional also has a critical point in this set. Then we apply this abstract theorem to the BEC system and obtain multiple sign-changing solutions, multiple mixed state of nodal solutions and one positive solution when the coupling parameter is positive but small. This result extends that of Chen, Lin and Zou about the existence of sign-changing solutions and semi sign-changing solutions, while our method and proof are totally di?erent. It also gives another kind of proof for the existence of a positive solution which is showed by Soave. When the BEC system satisfies some symmetric property, we construct multiple nontrivial solutions including sign-changing solutions of the system when the dimension N ≤ 4.Besides, we consider the general BEC related system with the critical exponent,i.e., the critical system with coupled terms. Through taking advantage of the manifold with two constraints as well as the mountain pass theorem, we obtain the result about the existence of a positive ground state solution. When the critical system possesses some symmetric property, by extending the result about the existence of solutions of the general Brezis-Nirenberg critical exponent problem, we can also construct multiple nontrivial solutions of the system. In addition, by priori estimates, we get some nonexistence result about the solutions of the system.Finally, we consider the BEC related perturbed symmetric system. Perturbed symmetric problem was proposed by A. Bahri, H. Berestycki, L. P. Lions, P. H. Rabinowitz,etc. in 1982, and it is still an open problem until now. For the system case, there is little result; while for the asymmetrical system with coupled terms, it has not been studied before as far as we can see. We construct an auxiliary problem and a sequence of modified minimax values which goes to infinity, then we show the existence of infinitely many solutions of the system.