Extremal solution and Liouville theorem for anisotropic elliptic equations

<p style='text-indent:20px;'>We study the quasilinear Dirichlet boundary problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases} \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda>0 $\end{document}</tex-math></inline-formula> is a parameter, id="M2">\begin{document}$ \Omega\subset\mathbb{R}^{N} (<inline-formula><tex-math id="M3">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula>) bounded domain, and operator id="M4">\begin{document}$ Q $\end{document}</tex-math></inline-formula>, known as Finsler-Laplacian or anisotropic Laplacian, defined by</p><p id="FE2"> Qu: \sum\limits_{i 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). style='text-indent:20px;'>Here, id="M5">\begin{document}$ F_{\xi_{i}} \frac{\partial F}{\partial\xi_{i}}(\xi) id="M6">\begin{document}$ F: \mathbb{R}^{N}\rightarrow [0, +\infty) convex function of id="M7">\begin{document}$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) satisfies certain assumptions. We derive existence extremal solution obtain that it regular, if id="M8">\begin{document}$ N\leq9 $\end{document}</tex-math></inline-formula>.</p><p also concern Hénon type Liouville equation, </p><p id="FE3"> (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N}, id="M9">\begin{document}$ \alpha>-2 id="M10">\begin{document}$ id="M11">\begin{document}$ F^{0} support id="M12">\begin{document}$ K: \{x\in\mathbb{R}^{N}:F(x)<1\} $\end{document}</tex-math></inline-formula>. theorem for stable solutions finite Morse index id="M13">\begin{document}$ 2\leq N<10+4\alpha id="M14">\begin{document}$ 3\leq N<10+4\alpha^{-} respectively, where id="M15">\begin{document}$ \alpha^{-} \min\{\alpha, 0\} $\end{document}</tex-math></inline-formula>.</p>