Classification of solutions to the singular Liouville's equation associated with the $N$ Finsler Laplacian

TL;DR

This paper classifies solutions to a singular Liouville equation involving the Finsler-$N$-Laplacian, extending previous results by relaxing the finite mass condition for solutions in $ ext{R}^N$.

Contribution

It provides a comprehensive classification of solutions to the Finsler-Liouville equation under broader conditions than prior work.

Findings

Classified solutions for the Finsler-Liouville equation with finite mass.

Extended the classification to cases with relaxed mass conditions.

Connected the solutions to the geometry of Finsler metrics.

Abstract

In this paper, we classify a class of singular Liouville's equation associated with the Finsler-$N$-Laplacian for any $\beta\in (0,N)$ \begin{align*} -\mathrm{div}\left(F^{N-1}(\nabla u)DF(\nabla u)\right)=\hat{F}^{o}(x)^{-\beta}e^u\ \ \text{in } \mathbb{R}^{N}\backslash \{0\}, \end{align*} under the finite mass condition $\int_{\mathbb{R}^{N}}\hat{F}^{o}(x)^{-\beta}e^u dx<+\infty$. Here $F$ is a convex function, which is positively homogeneous of degree 1, and its polar $F^{o}$ represents a Finsler metric on $\mathbb{R}^{N}$, $\hat{F}^{o}(x)=F^{o}(-x)$. Our result relaxes the mass condition required in the classification result in [39]