Extremal solution and Liouville theorem for anisotropic elliptic equations

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, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ with $N\ge 2$ be a bounded domain, and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by $$Qu:=\sum _{i=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}(F(\mathrm{\nabla }u)F_{{\xi }_i}(\mathrm{\nabla }u))\mathrm{.}$$ 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 $N\le 9$. We also concern the H'{e}non type anisotropic Liouville equation, namely, $$-Qu=(F^0(x){)}^{\alpha }{e}^u\text{quadin}{\mathbb{R}}^N$$ 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 $$F^0(x):=\underset{\xi \in K}{\sup }⟨x,\xi ⟩\mathrm{.}$$ We obtain the Liouville theorem for stable solutions and the finite Morse index solutions for $2\le N<10+4\alpha$ and $3\leq N<10+4\alpha^{-}$respectively,where$\alpha^{-}=\min{\alpha,0}$.