Extremal solution and Liouville theorem for anisotropic elliptic equations

We study the quasilinear Dirichlet boundary problem

$ \begin{equation} \nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u = 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases} \end{equation} $

where $ \lambda>0 $ is a parameter, $ \Omega\subset\mathbb{R}^{N} $ ($ N\geq2 $) is a bounded domain, and the operator $ Q $, known as Finsler-Laplacian or anisotropic Laplacian, is defined by

$ Qu: = \sum\limits_{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}}(\xi) $ and $ F: \mathbb{R}^{N}\rightarrow [0, +\infty) $ is a convex function of $ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $, and satisfies certain assumptions. We derive the existence of extremal solution and obtain that it is regular, if $ N\leq9 $.

We also concern the Hénon type anisotropic Liouville equation,

$ -Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\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\} $. We obtain the Liouville theorem for stable solutions and finite Morse index solutions for $ 2\leq N<10+4\alpha $ and $ 3\leq N<10+4\alpha^{-} $ respectively, where $ \alpha^{-} = \min\{\alpha, 0\} $.