Classification of solutions to the anisotropic $N$-Liouville equation in $\mathbb{R}^N$

TL;DR

This paper classifies all finite-mass solutions to the anisotropic N-Liouville equation involving the Finsler N-Laplacian in Euclidean space, confirming a conjecture for the two-dimensional case.

Contribution

It provides a complete classification of solutions to the anisotropic N-Liouville equation with finite mass, including a proof of a conjecture in the two-dimensional case.

Findings

Complete classification of solutions in 

Affirmative answer to the conjecture for N=2

Insights into anisotropic Liouville equations

Abstract

Given $N\geq 2$, we completely classify the solutions of the anisotropic $N$-Liouville equation $$-\Delta_N^H\,u=e^u \quad\text{in }\mathbb{R}^N,$$ under the finite mass condition $\int_{\mathbb{R}^N} e^u\,dx<+\infty$. Here $\Delta_N^H$ is the so-called Finsler $N$-Laplacian induced by a positively homogeneous function $H$. As a consequence for $N=2$, we give an affirmative answer to a conjecture made in [G. Wang and C. Xia, J. Differential Equations 252 (2012) 1668--1700].