Extremal solution and Liouville theorem for anisotropic elliptic equations

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 $\alpha^{-}=\min\{\alpha,0\}$.