Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Positive solutions to a supercritical elliptic problem which concentrate
  along a thin spherical hole   We consider the supercritical problem \[ -\Delta v=|v|^{p-2}v in
\Theta_{\epsilon}, v=0 on \partial\Theta_{\epsilon}, \] where $\Theta$ is a
bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $p>2^{\ast}:=2N/(N-2)$,
and $\Theta_{\epsilon}$ is obtained by deleting the $\epsilon$-neighborhood of
some sphere which is embedded in $\Theta$. In some particular situations we
show that, for $\epsilon>0$ small enough, this problem has a positive solution
$v_{\epsilon}$ and that these solutions concentrate and blow up along the
sphere as $\epsilon$ tends to 0.
  Our approach is to reduce this problem to a critical problem of the form \[
-\Delta u=Q(x)|u|^{4/(n-2)}u in \Omega_{\epsilon}, u=0 on
\partial\Omega_{\epsilon}, \] in a punctured domain
$\Omega_{\epsilon}:=\{x\in\Omega:|x-\xi_{0}|>\epsilon\}$ of lower dimension, by
means of some Hopf map. We show that, if $\Omega$ is a bounded smooth domain in
$\mathbb{R}^{n}$, $n\geq3$, $\xi_{0} is in\Omega,$ $Q is in
C^{2}(\b{\Oarmega})$ is positive and $\nabla Q(\xi_{0})\neq0$ then, for
$\epsilon>0$ small enough, this problem has a positive solution $u_{\epsilon}$,
and that these solutions concentrate and blow up at $\xi_{0}$ as $\epsilon$
goes to 0.

37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio