关于刘维尔型方程集中紧现象的研究

Liouville型方程的集中紧定理是半线性椭圆方程爆破分析的核心定理，是研究方程解的存在性的最重要的定理之一，因此Liouville型方程的集中紧现象的研究具有重要的理论价值和应用价值。. 本项目主要围绕Liouville型方程的集中紧现象及与之相关的问题展开研究。首先是展开分数阶Liouville型方程、Finsler-Liouville型方程以及高阶Liuoville型方程的研究，着重研究他们的集中紧定理和相关的爆破分析，并结合变分方法研究相应平均场方程解的存在性问题。其次是在前项目的基础上继续深入地研究Super-Liouville型方程组(包括带奇点的变系数方程及其Neumann/Chirality边值问题)的爆破分析和能量的量子性态。. 项目所涉及的方程类型丰富，解决问题的方法也不尽相同，我们希望以此完善半线性椭圆方程的理论。

One of the most important theorem of blow-up analysis for semi-linear elliptic equation is the concentration compactness theorem of Liouville-type equation. It is the crucial theorem to solve the equation. Therefore to develope the concentration compactness phenomena of Liouville-tpye equation is very valuable for theory and application. . This project will consider the corresponding problems on the concetration-compactness phenomena for several types of Liouville-type equation. Firstly, we will study Fractional-Liouville equation, Finsler-Liouville eqution and Hight-order-Liouville equation. We will develop their concentration-compactness theorem and blow-up analysis theory. We also will develop the existence theory of the corresponding mean field equation. Secondly, we will base on the former project to study the blow-up analysis for super-Liouville equations, including the blow-up anlysis for the variant coefficient super-Liouville equations with Neumann/Chirality boundary condition. . The project will study several types of Liouville-type equation. We will need new skill and new method to solve the problem. To this point, we hope we can perfect the theory of semi-linear elliptic equation.