Optimal concentration level of anisotropic Trudinger-Moser functionals on any bounded domain

Let $F$ be convex and homogeneous of degree $1$, its polar $F^o$ represent a finsler metric on ${\mathbb{R}}^n$, and $\mathrm{\Omega }$ be any bounded open set in ${\mathbb{R}}^n$. In this paper, we first construct the theoretical structure of anisotropic harmonic transplantation. Using the anisotropic harmonic transplantation, co-area formula, limiting Sobolev approximation method, delicate estimate of level set of Green function, we investigate the optimal concentration level of the Trudinger-Moser functional $${\int }_{\mathrm{\Omega }}{e}^{{\lambda }_n\mathrm{\mid }u{\mathrm{\mid }}^{\frac{n}{n-1}}}dx$$ under the anisotropic Dirichlet norm constraint ${\int }_{\mathrm{\Omega }}{F}^n\left(\mathrm{\nabla }u\right)dx\le 1$, where λ_{n}=n^{\frac{n}{n-1}}κ_{n}^{\frac{1}{n-1}}\ denotes the sharp constant of anisotropic Trudinger-Moser inequality in bounded domain and ${\kappa }_n$ is the Lebesgue measure of the unit Wulff ball. As an application. we can immediately deduce the existence of extremals for anisotropic Trudinger-Moser inequality on bounded domain. Finally, we also consider the optimal concentration level of the anisotropic singular Trudinger-Moser functional. The method is based on the limiting Hardy-Sobolev approximation method and constructing a suitable normalized anisotropic concentrating sequence.