由于Moser-Trudinger 不等式在调和分析和偏微分方程等学科中有广泛应用, 过去几十年, 人们做了大量的研究. 本文主要研究了Rⁿ中有界区域上带有L^p范数的奇异Moser-Trudinger 不等式及其极值函数的存在性.

      本文分为三个部分.第一部分, 我们通过构造测试函数, 举出反例, 得出了奇异Moser-Trudinger 不等式在α ≥ λ(?)时不成立.第二部分, 我们证明了在0 ≤ α < λ(?)时奇异的Moser-Trudinger 不等式成立.首先, 我们利用Adimurthih-Sandeep 带奇异项的集中紧原理证明了次临界的奇异Moser-Trudinger 不等式极值函数的存在性.其次, 我们建立了次临界情况下极值函数所满足的Euler-Lagrange 方程, 并对Euler-Lagrange 方程的解进行估计. 最后, 通过爆破分析对爆破点在区域内和区域边界两种情况分别讨论, 证明了临界情况下奇异的Moser-Trudinger 不等式成立. 第三部分, 我们借助Carleson-Chang 的方法给出集中紧序列对应的奇异Moser-Trudinger 不等式的上界估计, 并通过构造测试函数得到更大的下界估计, 推导出爆破序列不存在, 从而证明了临界情况下奇异Moser-Trudinger 不等式的极值函数存在.

The Moser-Trudinger inequality fascinates a lot of mathematicians and physicist due to their physical significance and application to harmonic analysis and different type of PDE’s, so for the past few decades, a lot of research has been done. In this dissertation, we mainly consider an singular Moser-Trudinger inequality with L p norm for bounded domains in R n and the existence of it’s extremal. This thesis is divided into three parts. Firstly, we construct test functions to obtain contradictions and show that the singular Moser-Trudinger inequality is infinity for α ≥ λ(Ω). Secondly, we prove that the singular Moser-Trudinger inequality is finite for 0 ≤ α < λ(Ω). In the first place, we use Adimurthi and Sandeep’s concentration compactness principle to show the existence of maximizers in subcritical case. Next, we construct the relevant Euler-Lagrange equation for the maximizers of the subcritical singular Moser-Trudinger function and investigate the asymptotic behavior of the maximizers. Then, we investigate the two case respectively through blow-up analysis. Thirdly, we use Carleson-Chang’s upper bound estimates to derive the upper bound estimates of singular Moser-Trudinger inequality with the blow-up sequence, and we get the lower bound through constructing test function, thereby the blow-up doesn’t exist, thus we prove the existence of the extremal functions of the improved singular Moser-Trudinger inequality.