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

Let $F:{\mathbb{R}}^n\to [0,+\mathrm{\infty })$ be a convex function of class $C^2({\mathbb{R}}^n\mathrm{\backslash }\{0\})$ 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 $${\int }_{{\mathbb{R}}^n}\mathrm{\Phi }\left({\lambda }_n{\mid u\mid }^{\frac{n}{n-1}}{(1+\alpha {\parallel u\parallel }_n^n)}^{\frac{1}{n-1}}\right)dx<+\mathrm{\infty }$$ in the entire space ${\mathbb{R}}^n$ for any $0\le \alpha <1$ is established, where $\Phi\left( t\right) =e^{t}-\underset{j=0}{\overset{n-2}{\sum}}% \frac{t^{j}}{j!}$,$\lambda_{n}=n^{\frac{n}{n-1}}\kappa_{n}^{\frac{1}{n-1}}$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 $\alpha \ge 1$. 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.