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

Given $N\ge 2$, we completely classify the solutions of the anisotropic $N$-Liouville equation $$-{\mathrm{\Delta }}_N^{H}u=e^u\text{in }{\mathbb{R}}^N,$$ under the finite mass condition ${\int }_{{\mathbb{R}}^N}{e}^{u}dx<+\mathrm{\infty }$. Here ${\mathrm{\Delta }}_N^H$ 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].