本项目的主要目的是指导访问学者对一类带非局部项的薛定谔方程组正解的存在性及其性态展开系统的研究。拟研究的问题来源于波色-爱因斯坦凝聚以及非线性光学。我们将运用传统的临界点理论与精致的分析技巧来探讨，当位势函数是两个相等的正常数时，方程组的极小能量解的非退化性以及正解的唯一性；同时，我们还将借助于Lyapunov-Schmidt约化方法探究，当位势函数满足一定的代数衰减时，方程组的同步、分离多峰解的存在性。通过研究上述问题，我们希望发展和开拓研究非局部问题的新方法，回答偏微分方程领域人们关心的一些问题，进一步促进我国西部偏微分方程的发展。因此，本项目的研究是十分有意义的。

The main purpose of this project is to guide a visiting scholar to study the existence and quantitative properties of positive solutions for a class of Schrödinger system with nonlocal terms, which originates from Bose-Einstein condensation and non-linear optics. We will try to use the traditional critical point theory and some delicate techniques from the nonlinear functional analysis to prove the non-degeneracy of the least energy solution and the uniqueness of the positive solutions for the given system when the potential functions in the system are equal positive constants. Meanwhile, we will also try to explore the existence of synchronized or separated multi-peak solutions for the system by Lyapunov-Schmidt reduction when the potential functions in the system satisfy some algebraic decay. It is possible to develop some new methods through the study of above Schrödinger system for some problems with nonlocal terms and answer some interesting problems in the field on Partial Differential Equations. Furthermore, our research also can promote the development of Partial Differential Equations in west part of China. Thus, our research is very important.