Extremal solution and Liouville theorem for anisotropic elliptic equations

TL;DR

This paper investigates extremal solutions and Liouville theorems for anisotropic elliptic equations involving the Finsler-Laplacian, establishing existence, regularity, and nonexistence results in various dimensions and conditions.

Contribution

It introduces new existence and regularity results for extremal solutions of anisotropic elliptic equations and extends Liouville theorems to anisotropic settings with stability and Morse index considerations.

Findings

Existence of regular extremal solutions for dimensions up to 9.

Liouville theorem for stable solutions in the anisotropic setting.

Finite Morse index solutions characterized for certain dimension and parameter ranges.

Abstract

We study the quasilinear Dirichlet boundary problem \begin{equation}\nonumber \left\{ \begin{aligned} -Qu&=\lambda e^{u} \quad \mbox{in}\quad\Omega\\ u&=0 \quad \mbox{on}\quad\partial\Omega,\\ \end{aligned} \right. \end{equation} where $\lambda>0$ is a parameter, $\Omega\subset\mathbb{R}^{N}$ with $N\geq2$ be a bounded domain, and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $$ Here, $F_{\xi_{i}}=\frac{\partial F}{\partial\xi_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $ C^{2}(\mathbb{R}^{N}\setminus\{0\})$, that satisfies certain assumptions. We derive the existence of extremal solution and obtain that it's regular, if $N\leq9$. We also concern the H\'{e}non type anisotropic Liouville equation, namely, $$-Qu=(F^{0}(x))^{\alpha}e^{u}\quad\mbox{in}\quad\mathbb{R}^{N}$$ where $\alpha>-2$, $N\geq2$ and $F^{0}$ is the support function of $K:=\{x\in\mathbb{R}^{N}:F(x)<1\}$ which is defined by $$F^{0}(x):=\sup_{\xi\in K}\langle x,\xi\rangle.$$ We obtain the Liouville theorem for stable solutions and the finite Morse index solutions for $2\leq N<10+4\alpha$ and $3\leq N<10+4\alpha^{-}$ respectively, where $\alpha^{-}=\min\{\alpha,0\}$.