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

Abstract

Let $F:{\mathbb{R}}^n\to [0,+\infty )$ 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 $0\le \alpha <1$ is established, where $\Phi \left(t\right)=e^t-\underset{j=0}{\sum ^{n-2}}$, ${\lambda }_n=n^{\frac{n}{n-1}}{\kappa }_n^{\frac{1}{n-1}}$ 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 $\alpha \ge 1$. 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.

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

References in corpus (1)

• Critical points of the Moser-Trudinger functional on a disk