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 $\left(\int_{\Omega}F^{N}(\nabla u)\;\mathrm{d}x\right)^{\frac{1}{N}}$ in the Sobolev-type space $D^{N,q}(\mathbb{R}^{N})$, $q\geq 1$, here $F:\mathbb{R}^{N}\rightarrow[0,+\infty)$ 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 $\|F(\nabla u)\|_{N}\leq 1$, the case $\|F(\nabla u)\|_{N}^{a}+\|u\|_{q}^{b}\leq 1$, and whether they are associated with exact growth.