微分几何和理论物理中很多重要问题的研究往往是转化为非线性椭圆方程的研究，即研究方程的解的存在性、唯一性和正则性等方程的分析问题。本项目主要采用这种几何分析的思想研究半线性椭圆方程理论及其几何与分析应用。具体地. 首先，我们将展开带奇点的半线性椭圆偏微分方程的blow-up分析、奇点的局部估计和相关问题解的存在性等方面的研究，进一步挖掘和创新非线性分析方法。. 其次，我们将展开带奇点的半线性椭圆方程的理论和方法的几何分析应用研究，比如对奇性曲面上的预定曲率问题提供分析估计、对奇性曲面上的一些几何和分析不等式提供理论依据。我们将着重研究奇性曲面上的一些几何问题,进一步完善几何分析理论。

Many important problems in Differential Geometry and Physics can be transfered some kinds of nonlinear elliptic euqation problem. So we need study the analytical properties of these nonlinear elliptic equation problems, i.e. we should study the existence, uniqueness and regularity of the euqtions. In this project we will study some semi-linear elliptic equations and its geometric and analytical applications. . First, we will consider some semi-linear elliptic equations with singular point, including the blow-up analysis, the local estimates near the singular point and the solution existence of corresponding equations. Therefore，we hope to get the new methods for the nonlinear anlysis.. Next, we will study the application of the thory of these semi-linear elliptic equations with singular point, for example, they will give the analytical estimates for prescribed curvature problem and some geometric and analytical inequalities. We will pay attention to studying some geometric problems on singular surface. We hope to perfect the theory of geometric analysis.