论文标题
各向异性Moser-Trudinger不平等涉及$ l^{n} $ norm在整个空间中$ \ mathbb {r}^{N} $
Anisotropic Moser-Trudinger inequality involving $L^{n}$ norm in the entire space $\mathbb{R}^{n}$
论文作者
论文摘要
令$ f：\ mathbb {r}^{n} \ rightarrow [0，+\ infty）$是类$ c^{2}（\ Mathbb {r}^{n} {n} \ backslash \ {0 \} $均匀且均匀的fin fir and porter的$ 1 $ 1的c^{2}（\ Mathbb {r}^{n} \ backslash \}） $ \ Mathbb {r}^{n} $上的度量。 $ w^{1，n} \ left（\ mathbb {r}^{n} \ right）$中的各向异性sobolev norm是由\ begin {equation*} || u || u || _ {f} = \ left（\ weft）定义的u）+| u |^{n} \ right）^{\ frac {1} {n}}。 \ end {equation*}在本文中，以下尖锐的各向异性Moser-Moser-trudinger不等式涉及$ l^{n} $ norm \ [\ inderSet {\ inset {u \ in w^{1，n}（\ mathbb {\ mathbb {r} 1} {\ sup} \ int_ {\ mathbb {r} ^{n}}φ\ left（λ_{λ_{n} \ left \ left \ vert \ vert u \ right \ vert ^{\ frac {\ frac {n} _ {n}^{n} \ right）^{\ frac {1} {n-1}}} \ right）dx <+\ fty \ infty \]在整个空间$ \ mathbb {r} = $λ_{n} = n^{\ frac {n} {n-1}}κ_{n}^{\ frac {\ frac {1} {n-1}} $和$κ__{n} $是单位wulff Ball的体积，是$ \ \ Mathbb {r} r}^n $。还表明，上述冠军是所有$α\ geq1 $的无穷大。此外，我们证明了至上的实现，即，当$α> 0 $足够小时，上述超级人物的最大化量是最大化器。本文的主要结果证明是基于爆炸分析方法。
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.
