Sharp anisotropic singular Trudinger-Moser inequalities in the entire space

In this paper, we investigate sharp singular Trudinger-Moser inequalities involving the anisotropic Dirichlet norm ${({\int }_{\mathrm{\Omega }}{F}^N(\mathrm{\nabla }u)\mathrm{d}x)}^{\frac{1}{N}}$ in the Sobolev-type space $D^{N,q}({\mathbb{R}}^N)$, $q\ge 1$, here $F:{\mathbb{R}}^N\to [0,+\mathrm{\infty })$ is a convex function of class $C^2({\mathbb{R}}^N\setminus \{0\})$, which is even and positively homogeneous of degree 1, its polar $F^0$ represents a Finsler metric on ${\mathbb{R}}^N$. Combing with the connection between convex symmetrization and Schwarz symmetrization, we will establish anisotropic singular Trudinger-Moser inequalities and discuss their sharpness under several different situations, including the case $\mathrm{\parallel }F(\mathrm{\nabla }u){\mathrm{\parallel }}_N\le 1$, the case $\mathrm{\parallel }F(\mathrm{\nabla }u){\mathrm{\parallel }}_N^a+\mathrm{\parallel }u{\mathrm{\parallel }}_q^b\le 1$, and whether they are associated with exact growth.