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