M-T (Moser Trudinger)不等式在偏微分方程、几何分析以及弦理论的研究中有着广泛的应用，是泛函分析领域的研究热点之一。本项目将重点关注Heisenberg群上M-T不等式的研究及其在次椭圆问题中的应用。.由于利用Heisenberg群上现有的M-T不等式不能得到具有指数临界增长的次椭圆问题基态解的存在性，本项目将对现有的M-T不等式进行改进，即，将研究Heisenberg群上与M-T不等式相关的集中紧性原理和具有精确增长的M-T不等式，此外，将结合变分法和临界点理论研究基态解的存在性等问题。.项目创新之处是：1) 在Heisenberg群上建立Talenti型比较原理，并用于研究M-T不等式；2) 将欧氏空间上椭圆问题基态解的存在性理论发展到次椭圆情形。.本项目的研究将有助于揭示Heisenberg群上M-T型不等式研究的一般规律，为相关次椭圆问题的进一步研究提供理论依据。

Due to the wide range of applications in partial defferential equations, geometric analysis and String theory, Moser-Trudinger type inequalities become one of the focus in the field of Functional analysis. In this project, we will be concerned with the study of the Moser Trudinger type inequalities on Heisenberg group and the related applications in Subelliptic problems..Since the existence of the ground state solutions to the subelliptic problems with critical exponential growth can not be obtained by the existing Moser-Trudinger inequalities on Heisenberg group, we will study the improved versions of these Moser-Trudinger inequalities, namely, the related Concentration Compactness principle and the Moser-Trudinger inequality with exact growth on the Heisenberg group, furthermore, we will apply the Variational method and Critical point theory to investigate the existence of the ground state solutions..The innovations are as follows: 1) Establishing the Talenti type comparison principle on the Heisenberg group, and use it to prove Moser-Trudinger type inequalities; 2) Generalizing the existence theory for ground state solutions to the elliptic problems in Euclidean space to the Subelliptic setting..This project will help to reveal the general rules for the studying of Moser-Trudinger inequalities on Heisenberg group, and provide the essential theoretical basis for the further study of related subelliptic equations.