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

Abstract

In this paper, we classify a class of singular Liouville&#39;s equation associated with the Finsler-$N$-Laplacian for any $β\in (0,N)$ \begin{align*} -\mathrm{div}\left(F^{N-1}(\nabla u)DF(\nabla u)\right)=\hat{F}^{o}(x)^{-β}e^u\ \ \text{in } \mathbb{R}^{N}\backslash \{0\}, \end{align*} under the finite mass condition $\int_{\mathbb{R}^{N}}\hat{F}^{o}(x)^{-β}e^u dx<+\infty$. 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]