Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations

TL;DR

This paper investigates regularity and Liouville theorems for stable solutions of anisotropic elliptic equations involving the Finsler-Laplacian, establishing bounds on singular set dimensions and proving nonexistence results in certain dimensions.

Contribution

It extends partial regularity results and Liouville theorems to anisotropic elliptic equations with the Finsler-Laplacian, including explicit solutions and power-type nonlinearities.

Findings

Hausdorff dimension of singular set ≤ N-10 for bounded domains

Liouville theorems for stable solutions in dimensions N<10

Explicit stable solutions outside compact sets in N=2

Abstract

We study the quasilinear elliptic equation \begin{equation*} -Qu=e^u \ \ \text{in} \ \ \Omega\subset \mathbb{R}^{N} \end{equation*} where the operator $Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)),$$ where $F_{\xi_{i}}=\frac{\partial F}{\partial\xi_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $ C^{2}(\mathbb{R}^{N}\setminus\{0\})$, that satisfies certain assumptions. For bounded domain $\Omega$ and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed $N-10$. For the entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions $N<10$ and $2<N<10$, respectively. We also provide an explicit solution that is stable outside a compact set in $N=2$. In addition, we provide similar Liouville theorems for the power-type nonlinearities.