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*} λ_{1}(Ω) = \inf_{u\in W_0^{1,n}(Ω),u\not\equiv 0} ||F(\nabla u)||_{L^n(Ω)}^n / ||u||_{L^n(Ω)}^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}(Ω),||F(\nabla u)||_{L^n(Ω)} = 1} \int_Ωe^{λ_n (1+α||u||_{L^n (Ω)}^n)^{\frac{1}{n-1}} |u|^{\frac{n}{n-1}}}dx \end{equation*} is finite for any $0\le \alpha <{\lambda }_1(\Omega )$,and the supremum is infinite for any $\alpha \ge {\lambda }_1(\Omega )$, 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 $0\le \alpha <{\lambda }_1(\Omega )$.