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