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'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)^{-β}e^u\ \ \text{in } \mathbb{R}^{N}\backslash \{0\}, \end{align*} under the finite mass condition ${\int }_{{\mathbb{R}}^N}{\hat{F}}^o(x{)}^{-\beta }{e}^{u}dx<+\infty$. 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]

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