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

Abstract

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 \[ \underset{u\in W^{1,n}( \mathbb{R}^{n}),\left\Vert u\right\Vert _{F}\leq 1}{\sup}\int_{ \mathbb{R} ^{n}}\Phi\left( \lambda_{n}\left\vert u\right\vert ^{\frac{n}{n-1}}\left( 1+\alpha\left\Vert u\right\Vert _{n}^{n}\right) ^{\frac{1}{n-1}}\right) dx<+\infty \] in the entire space $\mathbb{R}^n$ for any $0\leq\alpha<1$ is established, where $\Phi\left( t\right) =e^{t}-\underset{j=0}{\overset{n-2}{\sum}}% \frac{t^{j}}{j!}$, $\lambda_{n}=n^{\frac{n}{n-1}}\kappa_{n}^{\frac{1}{n-1}}$ and $\kappa_{n}$ is the volume of the unit Wulff ball in $\mathbb{R}^n$. It is also shown that the above supremum is infinity for all $\alpha\geq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $\alpha>0$ is sufficiently small. The proof of main results in this paper is based on the method of blow-up analysis.