各向异性Moser-Trudinger不等式及其应用研究

Moser-Trudinger不等式是Sobolev嵌入中一类非常重要的不等式，在几何分析和偏微分方程解的存在性方面有着广泛的应用。因此对Moser-Trudinger不等式及其应用的研究具有重要的理论价值和应用价值。. 本项目主要围绕Moser-Trudinger不等式及其相关的应用展开。首先是在前期的基础上展开对各向异性Moser-Trudinger不等式的进一步的研究，并同时研究此Moser-Trudinger嵌入极值函数是否存在。其次利用得到的各向异性Moser-Trudinger不等式并结合变分的方法展开对各向异性拉普拉斯平均场方程的解的存在性及其带临界指数的各向异性拉普拉斯方程正解存在性的研究。. 项目所涉及到的内容丰富，我们希望以此来丰富Sobolev嵌入等方面的理论。

The Moser-Trudinger is a class of very important inequality in the Sobolev embedding and can be widely used in the problem of the existence of the solution of the partial differential equation and the geometric analysis. Therefore, to develop the Moser-Trudinger inequality and its application is valuable for the theory and application.. This project will consider the corresponding problem on the Moser-Trudinger inequality.Firstly, we will base on the former work to study the extensive Moser-Trudinger inequality for anisotropic, especially, stress the emphasis on the existence of extremal function of the anisotropic Moser-Trudinger. Secondly, we will use this Moser-Trudinger inequality and variational method to study the existence of the mean field equation involving anisotropic Laplacian operator and the existence of the equation involving anisotropic operator with critical exponent growth.. The content of this project is very fruitful.To this point, we hope that we can perfect the theory of the Sobolev embedding.