Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A fountain of positive Bubbles on a Coron's Problem for a Competitive
  Weakly Coupled Gradient System   We consider the following critical elliptic system: \begin{equation*}
\begin{cases} -\Delta u_i=\mu_i u_i^{3}+\beta u_i^{ } \sum\limits_{j\neq i}
u_j^{2} \quad \hbox{in}\ \Omega_\varepsilon \\ u_i=0 \hbox{ on }
\partial\Omega_\varepsilon , \qquad u_i>0 \hbox{ in } \Omega_\varepsilon
\end{cases}\qquad i=1,\ldots, m, \end{equation*} in a domain
$\Omega_\varepsilon \subset \mathbb{R}^4$ with a small shrinking hole
$B_\varepsilon(\xi_0)$. For $\mu_i>0$, $\beta<0$, and $\varepsilon>0$ small, we
prove the existence of a non-synchronized solution which looks like a fountain
of positive bubbles, i.e. each component $u_i$ exhibits a towering blow-up
around $\xi_0$ as $\varepsilon \to 0$. The proof is based on the
Ljapunov-Schmidt reduction method, and the velocity of concentration of each
layer within a given tower is chosen in such a way that the interaction between
bubbles of different components balance the interaction of the first bubble of
each component with the boundary of the domain, and in addition is dominant
when compared with the interaction of two consecutive bubbles of the same
component.

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The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn