Anisotropic Finsler $N$-Laplacian Liouville equation in convex cones

Abstract

We consider the anisotropic Finsler $N$-Laplacian Liouville equation \[-Δ^{H}_{N}u=e^u \qquad {\rm{in}}\,\, \mathcal{C},\] where $N\ge 2$, $\mathcal{C}\subseteq {\mathbb{R}}^N$ is an open convex cone including ${\mathbb{R}}^N$, the half space ${\mathbb{R}}_{+}^N$ and $\frac{1}{2^m}$-space ${\mathbb{R}}_{2^{-m}}^N:=\{x\in {\mathbb{R}}^N\mid x_1,\cdots ,x_m>0\}$ ($m=1,\cdots ,N$), and the anisotropic Finsler $N$-Laplacian ${\Delta }_N^H$ is induced by a positively homogeneous function $H(x)$ of degree $1$. All solutions to the Finsler $N$-Laplacian Liouville equation with finite mass are completely classified. In particular, if $H(\xi )=|\xi |$, then the Finsler $N$-Laplacian ${\Delta }_N^H$ reduces to the regular $N$-Laplacian ${\Delta }_N$. Our result is a counterpart in the limiting case $p=N$ of the classification results in \cite{CFR} for the critical anisotropic $p$-Laplacian equations with $1<p<N$ in convex cones, and also extends the classification results in \cite{CK,CL,CW,CL2,E} for Liouville equation in the whole space ${\mathbb{R}}^N$ to general convex cones. In our proof, besides exploiting the anisotropic isoperimetric inequality inside convex cones, we have also proved and applied the radial Poincaré type inequality (Lemma \ref{A1}), which are key ingredients in the proof and of their own importance and interests.

38 pages, 1 figure