Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations

We study the quasilinear elliptic equation

$ \begin{equation*} -Qu = e^u ~~~~\mbox{in}~~~~ \Omega\subset \mathbb{R}^{N}, \end{equation*} $

where the operator $ Q $, known as the Finsler-Laplacian (or anisotropic Laplacian) operator, 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}} $ and $ F: \mathbb{R}^{N}\rightarrow[0, +\infty) $ is a convex function of $ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $ that satisfies certain assumptions. For a bounded domain $ \Omega $ and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed $ N-10 $. For the case of entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions $ N<10 $ and $ 2<N<10 $, respectively. We also provide an explicit solution that is stable outside a compact set in two dimensions $ N = 2 $. In addition, we present similar Liouville theorems for the related equations with power-type nonlinearities.