1960年H.Yamabe提出Yamabe问题，使之逐渐成为几何和分析中研究的重要内容之一，经过20多年的努力，最终由R.Schoen于1984年解决。而Yamabe方程解集的紧性也由R.Schoen、S.Brendle、F.C.Marques等数学家到2009年全部解决。随着完全非线性偏微分方程理论的发展，特别是Monge-Ampere方程和k-Hessian方程的发展，经典Yamabe方程逐渐推广到完全非线性的$\sigma_k$-Yamabe方程，但多数工作都集中在紧致无边流形上的研究。本项目旨在研究有界区域上的$\sigma_2$-Yamabe方程，综合利用极值原理、分部积分和爆破分析等方法和技巧来刻画$\sigma_2$-Yamabe方程的几何与分析，其内容涵盖Brezis-Nirenberg问题的推广、自然变分问题、解的几何形状和性质以及解集的紧性等。

Since H. Yamabe put forward the Yamabe problem in 1960, it has gradually become one of the important research contents in geometry and analysis. After more than 20 years’ efforts, it was finally settled by R. Schoen in 1984. Up to 2009, the problem about compactness of solution sets for the Yamabe equation was also completely solved by mathematicians such as R. Schoen, S. Brendle, F.C. Marques, etc. With the development of the theory of fully nonlinear partial differential equations, especially the Monge-Ampere equation and the k-Hessian equation, the classical Yamabe equation is gradually extended to the fully nonlinear $\sigma_k$-Yamabe equation. But most of the work on this equation is focused on compact manifolds without boundary. The purpose of this project is to study $\sigma_2$-Yamabe equation in the bounded domain. We research the geometry and analysis of the $\sigma_2$-Yamabe equation by methods and techniques mainly concerning maximum principle, integration by parts, and blow-up analysis. The contents of this project cover the generalization of the Brezis-Nirenberg problem, natural variational problems, geometric shapes and properties of solutions, compactness of solution sets, and so on.