Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the regularity and partial regularity of extremal solutions of a
  Lane-Emden system   In this paper, we consider the system $-\Delta u =\lambda (v+1)^p,\;\;-\Delta
v = \gamma (u+1)^\theta$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^N$
with the Dirichlet boundary condition $u=v=0$ on $\partial \Omega.$ Here $
\lambda,\gamma$ are positive parameters. Let $x_0$ be the largest root of the
polynomial \begin{equation*} H(x) = x^4 -
\frac{16p\theta(p+1)(\theta+1)}{(p\theta-1)^2}x^2 +
\frac{16p\theta(p+1)(\theta+1)(p+\theta+2)}{(p\theta-1)^3}x
-\frac{16p\theta(p+1)^2(\theta+1)^2}{(p\theta-1)^4}. \end{equation*} We show
that the extremal solutions associated to the above system are bounded provided
$N<2+2x_0.$ This improves the previous work in \cite{co1}. We also prove that,
if $N\geq 2+2x_0,$ then the singular set of any extremal solution has Hausdorff
dimension less or equal to $N-(2+2x_0).$

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The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci