Classification of Solutions to the Anisotropic N-Liouville Equation in ℝN

Abstract

Abstract Given $N\geq 2$, we completely classify solutions to the anisotropic $N$-Liouville equation $$ \begin{align*} &-\Delta_N^H\,u=e^u \quad\textrm{in}\ \mathbb{R}^N,\end{align*} $$ 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 in the planar case $N=2$, we give an affirmative answer to a conjecture made in [ 53].