Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Partial regularity and Liouville theorems for stable solutions of
  anisotropic elliptic equations   We study the quasilinear elliptic equation \begin{equation*} -Qu=e^u \ \
\text{in} \ \ \Omega\subset \mathbb{R}^{N} \end{equation*} where the operator
$Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by
$$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla
u)F_{\xi_{i}}(\nabla u)),$$ where $F_{\xi_{i}}=\frac{\partial
F}{\partial\xi_{i}}$ 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 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 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 $N=2$. In addition, we provide
similar Liouville theorems for the power-type nonlinearities.

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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys