Sharp anisotropic singular Trudinger-Moser inequalities in the entire space

TL;DR

This paper establishes sharp anisotropic singular Trudinger-Moser inequalities in the entire space, involving convex Finsler metrics and exploring their sharpness under various norm constraints.

Contribution

It introduces new sharp anisotropic inequalities involving Finsler metrics and analyzes their optimality in different norm settings.

Findings

Established anisotropic singular Trudinger-Moser inequalities.

Proved the sharpness of these inequalities under multiple conditions.

Connected convex symmetrization with Schwarz symmetrization for the proofs.

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.