Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on a sinh-Poisson type equation with variable intensities on
  pierced domains   We consider a sinh-Poisson type equation with variable intensities and
Dirichlet boundary condition on a pierced domain \begin{equation*} \left\{
\begin{array}{ll} \Delta u +\rho\left(V_1(x)e^{u}- V_2(x)e^{-\tau u}\right)=0
&\text{in } \Omega_\epsilon:=\Omega\setminus \displaystyle \bigcup_{i=1}^m
\overline{B(\xi_i,\epsilon_i)}\\ u=0&\text{on }\partial\Omega_\epsilon,
\end{array}\right. \end{equation*} where $\rho>0$, $V_1,V_2>0$ are smooth
potentials in $\Omega$, $\tau>0$, $\Omega$ is a smooth bounded domain in
$\mathbb{R}^2$ and $B(\xi_i,\epsilon_i)$ is a ball centered at $\xi_i\in
\Omega$ with radius $\epsilon_i>0$, $i=1,\dots,m$. When $\rho>0$ is small
enough and $m_1\in \{1,\dots,m-1\}$, there exist radii
$\epsilon=(\epsilon_1,\dots,\epsilon_m)$ small enough such that the problem has
a solution which blows-up positively at the points $\xi_1,\dots,\xi_{m_1}$ and
negatively at the points $\xi_{m_1+1},\dots,\xi_{m}$ as $\rho\to 0$. The result
remains true in cases $m_1=0$ with $V_1\equiv 0$ and $m_1=m$ with $V_2\equiv
0$, which are Liouville type equations.

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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: ao-sci