非局部预定曲率问题和非局部椭圆问题的变分方法

近年来，在物理和几何相关的问题研究中，非局部椭圆算子相关的问题研究受到了越来越多数学家的关注，这不仅在于其重要的理论意义，还在于它的应用价值。本项目研究的目的之一是利用变分法结合爆破分析技巧、Moser理论等工具研究在几何分析领域的热点问题分数阶Yamabe问题、分数阶Nirenberg类型问题、分数阶最优Sobolev不等式以及分数阶预定Morse数量曲率问题。另外，在本项目中我们希望利用伪指标理论结合Nehari流形方法研究来自物理相关问题的非局部椭圆问题，比如分数阶Schrodinger方程，分数阶Choquard方程以及分数阶Schrodinger-Choquard方程等问题多解的存在性及渐近行为。这些非局部问题都是物理和几何问题研究中热点问题之一。

In recent years, in the research of physics and geometry related problems, more and more mathematicians pay attention to the research of the problems related to nonlocal elliptic operators, which not only lies in its important theoretical significance, but also in its practical application value. One of the purposes of this project is to study the hot problems in the field of geometric analysis, such as fractional Yamabe problem, fractional Nirenberg type problem, fractional optimal Sobolev inequality and fractional prescribed Morse scalar curvature problem by using variational method and combining with the blow up analysis technique、 Moser theory and other tools. In addition, in this project, we also want to use the pseudo index theory and Nehari manifold method to study the existence and asymptotic behavior of multiple solutions of nonlocal elliptic problems from physical related problems, such as fractional Schrodinger equation, fractional Choquard equation and fractional Schrodinger-Choquard equation. These nonlocal problems are all hot issues in physics and geometry.