A sharp Trudinger-Moser type inequality involving L-n norm in the entire space R-n
A sharp Trudinger-Moser type inequality involving L-n norm in the entire space R-n
复制标题
涉及整个空间 R-n 中的 L-n 范数的尖锐 Trudinger-Moser 型不等式
DOI：
10.1016/j.jde.2019.03.037
复制
发表时间：
2019
影响因子：
2.4
通讯作者：
Zhu Maochun
中科院分区：
文献类型：
--
作者：
Lu Guozhen;Zhu Maochun
设W1，n（Rn）是标准Sobolev空间，W1，n（Rn）是Rn上的Ln范数.本文建立了如下Trudinger-Moser不等式的一个精确形式，它涉及Ln模的上界<$u <$W1，n（Rn）= 1 <$Rn Φ（α n| u| n n− 1（1+ α <$u <$n n）1 n− 1）dx <+∞ for any 0≤ α< 1，where Φ（t）= e t−∑ n− 2 j= 0 t j j！，α n= n ω n− 1 1 n− 1和ω n− 1是Rn中单位球的n− 1维曲面测度。我们还证明了对于所有α≥ 1，上述上确界是无穷大的。证明了当α> 0充分小时，上确界是存在的，即上确界存在极大值.证明基于Trudinger-Moser泛函的非线性Euler-Lagrange方程的爆破分析方法。我们的结果强化了最近的工作[19]，其中他们表明，当Φ（t）被严格较小的Φ（t）= et-∑ n-1 j= 0 t j j时，上述不等式以较弱的形式成立！(note（1）（2）（1）（
Let W 1, n (R n) be the standard Sobolev space and‖⋅‖ n be the L n norm on R n. We establish a sharp form of the following Trudinger-Moser inequality involving the L n norm sup‖ u‖ W 1, n (R n)= 1∫ R n Φ (α n| u| n n− 1 (1+ α‖ u‖ n n) 1 n− 1) d x<+∞ for any 0≤ α< 1, where Φ (t)= e t−∑ n− 2 j= 0 t j j!, α n= n ω n− 1 1 n− 1 and ω n− 1 is the n− 1 dimensional surface measure of the unit ball in R n. We also show that the above supremum is infinity for all α≥ 1. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when α> 0 is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals. Our results sharpen the recent work [19] in which they show that the above inequality holds in a weaker form when Φ (t) is replaced by a strictly smaller Φ⁎(t)= e t−∑ n− 1 j= 0 t j j!(note that Φ (t)= Φ⁎(t)+ t n− 1 (n− 1)!).