Sharp anisotropic singular Trudinger-Moser inequalities in the entire space

Abstract

In this paper, we investigate sharp singular Trudinger-Moser inequalities involving the anisotropic Dirichlet norm ${\left({\int }_{\hat{I}}\text{c◯}{F}^N(\nabla u)\mathrm{d}x\right)}^{\frac{1}{N}}$ in the Sobolev-type space $D^{N,q}({\mathbb{R}}^N)$, $q\ge 1$, here $F:{\mathbb{R}}^N\to [0,+\infty )$ 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 $\Vert F(\nabla u){\Vert }_N\le 1$, the case $\Vert F(\nabla u){\Vert }_N^a+\Vert u{\Vert }_q^b\le 1$, and whether they are associated with exact growth.

Anisotropic and singularity; Trudinger-Moser inequalities; Convex symmetrization; Sharp constants