Extremal solution and Liouville theorem for anisotropic elliptic equations

Abstract

We study the quasilinear Dirichlet boundary problem \begin{equation}\nonumber \left\{ \begin{aligned} -Qu&=λe^{u} \quad \mbox{in}\quadΩ\\ u&=0 \quad \mbox{on}\quad\partialΩ,\\ \end{aligned} \right. \end{equation} where $\lambda >0$ is a parameter, $\Omega \subset {\mathbb{R}}^N$ with $N\ge 2$ 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 $F:{\mathbb{R}}^N\to [0,+\infty )$ 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 $N\le 9$. We also concern the Hénon type anisotropic Liouville equation, namely, $$-Qu=(F^{0}(x))^αe^{u}\quad\mbox{in}\quad\mathbb{R}^{N}$$ where $\alpha >-2$, $N\ge 2$ and $F^0$ is the support function of $K:=\{x\in {\mathbb{R}}^N:F(x)<1\}$ which is defined by $$F^0(x):=\underset{\xi \in K}{\sup }⟨x,\xi ⟩.$$ We obtain the Liouville theorem for stable solutions and the finite Morse index solutions for $2\le N<10+4\alpha$ and $3\le N<10+4{\alpha }^{-}$ respectively, where ${\alpha }^{-}=\min \{\alpha ,0\}$.

References in corpus (1)

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