变分方法在椭圆型方程的研究中发挥着巨大的作用。本项目拟研究具有变分结构的四类Bose-Einstein方程组问题：第一类是权函数对Bose-Einstein方程组解的存在性的影响；第二类是含临界指标Bose-Einstein方程组问题；第三类是区域对Bose-Einstein方程组解的存在性的影响；第四类是分数阶Bose-Einstein方程组。我们拟采用Lyapunov-Schmidt约化方法，并结合局部Pohozaev恒等式研究这四类问题解的性态及解的局部唯一性。众所周知，方程组中分量之间的相互作用使其与单个方程区别很大。因此，对Bose-Einstein方程组的研究，有利于我们进一步揭示方程组与单个方程之间的共性和个性。

Variational method plays a key role in the study of elliptic equations. In this project, we aim to study four problems for Bose-Einstein systems with variational framework. The first one is the effect of potentials on the existence of solutions to Bose-Einstein systems. The second one is the Bose-Einstein systems with critical exponents. The third one is the effect of domains on the existence of solutions to Bose-Einstein systems. The fourth one is fractional Bose-Einstein systems. We will use Lyapunov-Schmidt reduction method, together with the local Pohozeav identities, to study the properties and the local uniqueness for solutions to the above four problems. It is well known that there are striking differences between systems and single equation due to the interaction of different components in the systems. Therefore, by studying Bose-Einstein systems, it will help us to reveal both the similarities and the differences between systems and single equations.