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arXiv 2005.06513math.AP
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全空间$\mathbb{R}^{n}$中涉及$L^{n}$范数的各向异性Moser-Trudinger不等式

Anisotropic Moser-Trudinger inequality involving $L^{n}$ norm in the entire space $\mathbb{R}^{n}$

Rulong Xie

备注：
arXiv admin note: substantial text overlap with arXiv:1703.00901, arXiv:1904.10531 by other authors
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摘要：

设$F: \mathbb{R}^{n}\rightarrow [0,+\infty) $是$C^{2}( \mathbb{R}^{n}\backslash\{0\})$类的一个凸函数，它是一个1次正齐次偶函数，它的极函数$F^{0}$表示$\mathbb{R}^{n}$上的一个Finsler度量. $W^{1,n}\left(\mathbb{R}^{n}\right)$中的各向异性Sobolev范数由\begin{equation*} ||u||_{F}=\left(\int_{\mathbb{R}^{n}}F^{n}(\nabla u)+|u|^{n}\right)^{\frac{1}{n}}. \end{equation*}定义本文对任意的$0\leqα<1$，在整个空间$\mathbb{R}^n$中建立了包含$L^{n}$范数\[ \underset{u\in W^{1,n}( \mathbb{R}^{n}),\left\Vert u\right\Vert _{F}\leq 1}{\sup}\int_{ \mathbb{R} ^{n}}Φ\left( λ_{n}\left\vert u\right\vert ^{\frac{n}{n-1}}\left( 1+α\left\Vert u\right\Vert _{n}^{n}\right) ^{\frac{1}{n-1}}\right) dx<+\infty \]的各向异性Moser-Trudinger不等式，其中$Φ\left( t\right) =e^{t}-\underset{j=0}{\overset{n-2}{\sum}}% \frac{t^{j}}{j!}$，$λ_{n}=n^{\frac{n}{n-1}}κ_{n}^{\frac{1}{n-1}}$，$κ_{n}$是$\mathbb{R}^n$中单位Wulff球的体积.它还表明，上述上确界是无穷大的所有$α\geq1$。证明了当$α>0$足够小时，上确界是存在的，即存在一个极大化子。本文主要结果的证明是基于爆破分析的方法。

英文摘要：

Let $F: \mathbb{R}^{n}\rightarrow [0,+\infty) $ be a convex function of class $C^{2}( \mathbb{R}^{n}\backslash\{0\})$ which is even and positively homogeneous of degree 1, and its polar $F^{0}$ represents a Finsler metric on $\mathbb{R}^{n}$. The anisotropic Sobolev norm in $W^{1,n}\left(\mathbb{R}^{n}\right)$ is defined by \begin{equation*} ||u||_{F}=\left(\int_{\mathbb{R}^{n}}F^{n}(\nabla u)+|u|^{n}\right)^{\frac{1}{n}}. \end{equation*} In this paper, the following sharp anisotropic Moser-Trudinger inequality involving $L^{n}$ norm \[ \underset{u\in W^{1,n}( \mathbb{R}^{n}),\left\Vert u\right\Vert _{F}\leq 1}{\sup}\int_{ \mathbb{R} ^{n}}Φ\left( λ_{n}\left\vert u\right\vert ^{\frac{n}{n-1}}\left( 1+α\left\Vert u\right\Vert _{n}^{n}\right) ^{\frac{1}{n-1}}\right) dx<+\infty \] in the entire space $\mathbb{R}^n$ for any $0\leqα<1$ is established, where $Φ\left( t\right) =e^{t}-\underset{j=0}{\overset{n-2}{\sum}}% \frac{t^{j}}{j!}$, $λ_{n}=n^{\frac{n}{n-1}}κ_{n}^{\frac{1}{n-1}}$ and $κ_{n}$ is the volume of the unit Wulff ball in $\mathbb{R}^n$. It is also shown that the above supremum is infinity for all $α\geq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $α>0$ is sufficiently small. The proof of main results in this paper is based on the method of blow-up analysis.

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