Moser-Trudinger-Adams不等式在几何分析，数学物理以及弦理论等研究中都有着广泛的应用，是泛函分析领域研究的热点问题之一。本项目将使用非线性泛函分析的工具研究全空间上Adams不等式极值函数的存在性以及与此相关的椭圆方程量化性质的问题。. 由于现有的对称重排技术不适用于研究全空间上Adams不等式极值函数的存在性问题，本项目拟寻求替代对称重排技术的新方法来研究全空间上Adams不等式极值函数的存在性。此外，本项目也将利用爆破分析和临界点理论等工具对四维空间上的单位球和全空间上带有指数增长的高阶方程的量化性质进行深入研究。. 本项目的研究将有助于揭示全空间上Moser-Trudinger-Adams极值函数存在性的一般规律，为进一步研究Adams泛函临界点相关问题提供了一定的理论基础。

Due to the wide range of applications in mathematical physics, geometric analysis and string theory, Moser-Trudinger-Adams inequalities have become one of the focus in the field of functional analysis. In this project, we will employ the method of nonlinear functional analysis to study the existence of extremals for the Adams inequalities on the whole space and the quantization property for the related elliptic equqtions. . Since the existence of extremals for the Adams inequalities on the whole space can't be obtained by the existing technique of the symmetry and rearrangement, we need find a new method to investigate the existence of the extremals for the Adams inequality on the whole space. Furthermore, we also will employ the methods combining blow-up analysis and critical point theory to establish the quantization property for high-order elliptic equations related with the extremals of the Adams inequalities on the unit ball and whole space. . The study of this project will contribute to reveal the general rules for the existence of extremals for the Moser-Trudinger-Adams inequalities on the whole space and provide the essential theoretical basis for the study of problems of the critical point related with the Adams functional. Furthermore, the study of this project will also provide new ideas for the further research of the extremals related with the Moser-Trudinger-Adams inequalities on non-compact manifolds.