本项目研究分数阶Yamabe方程及其相关问题：(1)研究黎曼流形和CR流形上的分数阶Yamabe问题，利用无穷远处临界点理论得到正解的存在性和多重性；(2)研究分数阶Yamabe流，利用Lyapunov-Schmidt约化方法构造其不同于收缩球解和King解的第II型古典解，揭示分数阶Yamabe流本质上不同于二维Ricci流的现象；(3)证明分数阶Choquard方程基态解的非退化性并研究多包解的存在性。

Our aim is to study the fractional Yamabe equation and its related topics: (1)Using critical point theory at infinity we study the fractional Yamabe problem, give a new proof of Gonzalez and Jie Qing's result, obtain the existence of positive solutions on Riemannian and CR manifolds. (2)We construct type II ancient solutions which are different from the contraction sphere solutions and King's solutions for the fractional Yamabe flow using Lyapunov-Schmidt reduction method. This reveals an essential difference between fractional Yamabe flow and two dimensional Ricci flow. (3)We prove the nondegeneracy of the ground state solutions for the fractional Choquard equation and the existence of multi-bump solutions.