本项目计划研究与Bose-Einstein 凝聚相关的非线性薛定谔方程组，包括三方面：一是Sirakov 2007年提出的公开问题，拟解决正解的唯一存在性，了解方程组解的基本性态。非对称情形下， 找出极小能量解存在和不存在的最佳耦合参数的范围，完全解决这个Open问题。二是高维（临界）情形的Bose-Einstein 凝聚型方程组， 解决正解的唯一存在性、找出极小能量的精确表达式，解决无穷多变号解的存在性、相位分离等问题。 三是非线性薛定谔方程组的奇异扰动问题， 研究当非线性项和Bose-Einstein 凝聚方程右端项相同时（包括低维数和高维数（临界）），基态解的存在性与Plank常数的关系。 研究Ambrosetti 和Malchiodi 2010年提出的Open问题。本项目所选研究主题， 都是当前本领域国际前沿热点问题，有十分重要的科学意义。 通过该项目， 推动变分法及其应用的发展。

We study the Bose-Einstein condensates and the related Schrodinger equations or systems. Firstly, we plan to solve the Sirakov's open problems raised in 2007, we will establish the unique existence on the positive solutions and obtain the fundamental property of the ground states. Without symmetric assumption, we shall find the optimal range of the coupling constant about the existence or nonexistence of the leat energy solutions, find the asymptotic behaviors of the ground states as the parameters change. Secondly, we study the higher dimensional and critical Bose-Einstein condenstaes systems, including the existence of the positive solutions, the precise formulas for the least energy in terms of the parameters, we will establish the existence of the infinitely many sign-changing solutions and determine the phase separation. Thirdly, we study the singularly perturbed Schrodinger systems, find the existence of the positive solutions and the condensation behaviors. Also, we will study the Ambrosetti-Malchiodi's open questions proposed in 2010, we mainly focus on the critical case. These topics are rather important and popular, they lie in the frontier of the modern mathematical researches. This project will absolutely move forward and advance the development of the variational and topological methods with applications in PDEs.