Anisotropic Moser-Trudinger inequality involving $L^n$ norm

Abstract: The paper is concerned about a sharp form of Anisotropic Moser-Trudinger inequality which involves $L^{n}$ norm. Let \begin{equation*} \lambda_{1}(\Omega) = \inf_{u\in W_0^{1,n}(\Omega),u\not\equiv 0} ||F(\nabla u)||{L^n(\Omega)}^n / ||u||{L^n(\Omega)}^n \end{equation*} be the first eigenvalue associated with $n$-Finsler-Laplacian. using blowing up analysis, we obtain that \begin{equation*} \sup_{u\in W_{0}^{1,n}(\Omega),||F(\nabla u)||{L^n(\Omega)} = 1} \int{\Omega}e^{\lambda_n (1+\alpha||u||{L^n (\Omega)}^n)^{\frac{1}{n-1}} |u|^{\frac{n}{n-1}}}dx \end{equation*} is finite for any $0\leq \alpha<\lambda{1}(\Omega)$,and the supremum is infinite for any $\alpha\geq \lambda_{1}(\Omega)$, where $\lambda_{n} = n^{\frac{n}{n-1}} \kappa_n^{\frac{1}{n-1}}$ ($\kappa_{n}$ is the volume of the unit wulff ball) and the function $F$ is positive,convex and homogeneous of degree $1$, and its polar $F^o$ represents a Finsler metric on $\mathbb{R}^n$. Furthermore, the supremum is attained for any $0\leq \alpha<\lambda_{1}(\Omega)$.