发表机构
Dalian University of Technology; DUT-BSU Joint Institute, Dalian University of Technology（大连理工大学; 大连理工大学-北京工业大学联合学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用集中紧性原理证明在标准Sobolev范数下Moser-Trudinger型不等式的上确界一致有界，并证明极值的存在性及极值序列在C^1中的紧性。
AI 中文摘要
本文利用集中紧性原理证明：若将Dirichlet范数替换为标准Sobolev范数，则对所有此类函数，积分$$ \int_{\mathbb{B}} |x|^{N \epsilon} \Phi \left( \alpha_N \left( 1+ \epsilon \right) |u|^{ \frac{N}{N-1} } \right) dx $$的上确界一致有界。此外，我们还证明了极值的存在性。最后，我们考虑不等式极值序列的紧性，该序列的极限是积分$$ \int_{\mathbb{B}} \Phi \left( \alpha_N |u|^{ \frac{N}{N-1} } \right) dx $$在$C^1 \left( \mathbb{B} \right)$中的极值，其中$\alpha_N = N \omega_{N-1}^{ \frac{1}{N-1} } $，$\Phi \left( t \right):= e^t - \sum_{j=0}^{N-2} \frac{t^j}{j!} $，$\omega_{N-1}$是$\mathbb{R}^N$中单位球的表面积。
英文摘要
In this paper, we employ the concentration-compactness principle to show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of $$ \int_{\mathbb{B}} |x|^{N ε} Φ\left( α_N \left( 1+ ε\right) |u|^{ \frac{N}{N-1} } \right) dx $$ over all such functions is uniformly bounded. Furthermore, we also prove the existence of extremals. Finally, we consider the compactness of the sequence of extremals of the inequalities and the limit of this sequence is the extremal of $$ \int_{\mathbb{B}} Φ\left( α_N |u|^{ \frac{N}{N-1} } \right) dx $$ in $C^1 \left( \mathbb{B} \right)$, where $α_N = N ω_{N-1}^{ \frac{1}{N-1} } $, $Φ\left( t \right) := e^t - \sum_{j=0}^{N-2} \frac{t^j}{j!} $ and $ω_{N-1}$ is the surface of the unit ball in $\mathbb{R}^N$.
