具有指数Neumann边界条件的Liouville型方程解的存在性

对微分几何和数学物理中的很多问题的研究，往往是转化为相应Liouville型方程的研究。在本项目中，我们基于共形黎曼几何的背景，考虑一类具有指数Neumann边界条件的Liouville型方程。此类方程具有共形不变特征，在几何中具有重要的意义。我们将研究边界具有锥形奇点的Riemann流形上的边界Moser-Trudinger不等式，继而结合变分法以及爆破分析的结果，讨论参数位于临界以及超临界情形时，方程解的存在性问题。通过本项目，我们期望更深入的理解有边Riemann流形上的共形度量存在性问题，进一步的完善半线性椭圆型方程理论体系。

The research of many problems in differential geometry and mathematical physics is often transformed into the research of corresponding Liouville type equation. In this project, based on the background of conformal Riemann geometry, we consider a class of Liouville type equation with exponential Neumann boundary condition. This kind of equation has the conformal invariable characteristic and important significance in geometry. We will study the boundary Moser-Trudinger inequality on Riemann manifold with conical singularities and discuss the existence of solution when the parameters of the equation are critical and supercritical by using variational method and blow up analysis. Through this project, we expect to have a deeper understanding of the existence of conformal metric on Riemann manifold and further improve the theoretical system of semilinear elliptic equation.