arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1801.03271math.AP
收藏

非齐次约束下Moser-Trudinger型不等式极大值解的存在性与不存在性

Existence and non-existence of maximizers for the Moser-Trudinger type inequalities under inhomogeneous constraints

Norihisa Ikoma, Michinori Ishiwata, Hidemitsu Wadade

备注：
Mathematische Annalen, 2019
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文研究了$\Bbb R^N$中形如\[ D_{N,α}(a,b):= \sup_{u\in W^{1,N}(\Bbb R^N),\,\|\nabla u\|_{L^N(\Bbb R^N)}^a+\|u\|_{L^N(\Bbb R^N)}^b=1} \int_{\Bbb R^N}Φ_N\left(α|u|^{N'}\right)dx. \]的Moser-Trudinger型不等式的极大化解的存在性和不存在性，这里$N\geq 2$，$N'=\frac{N}{N-1}$，$a,b>0$，$α\in (0,α_N]$和$Φ_N(t):=e^t-\sum_{j=0}^{N-2}\frac{t^j}{j!}$，其中$α_N:= N ω_{N-1}^{1/(N-1)}$和$ω_{N-1}$表示$\Bbb R^N$中单位球的表面积.我们表明存在的阈值$α_\ast = α_\ast(a,b,N) \in [0,α_N]$，使$D_{N,α}(a,b)$不能达到，如果$α\in (0,α_\ast)$和达到，如果$ α\in (α_\ast , α_N)$。我们还给出了不等式$α_\ast < α_N$成立的条件.

英文摘要：

In this paper, we study the existence and non-existence of maximizers for the Moser-Trudinger type inequalities in $\Bbb R^N$ of the form \[ D_{N,α}(a,b):= \sup_{u\in W^{1,N}(\Bbb R^N),\,\|\nabla u\|_{L^N(\Bbb R^N)}^a+\|u\|_{L^N(\Bbb R^N)}^b=1} \int_{\Bbb R^N}Φ_N\left(α|u|^{N'}\right)dx. \] Here $N\geq 2$, $N'=\frac{N}{N-1}$, $a,b>0$, $α\in (0,α_N]$ and $Φ_N(t):=e^t-\sum_{j=0}^{N-2}\frac{t^j}{j!}$ where $α_N:= N ω_{N-1}^{1/(N-1)}$ and $ω_{N-1}$ denotes the surface area of the unit ball in $\Bbb R^N$. We show the existence of the threshold $α_\ast = α_\ast(a,b,N) \in [0,α_N]$ such that $D_{N,α}(a,b)$ is not attained if $α\in (0,α_\ast)$ and is attained if $ α\in (α_\ast , α_N)$. We also provide the conditions on $(a,b)$ in order that the inequality $α_\ast < α_N$ holds.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑