Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Boundary clustered layers near the higher critical exponents   We consider the supercritical problem {equation*} -\Delta u=|u|
^{p-2}u\text{\in}\Omega,\quad u=0\text{\on}\partial\Omega, {equation*} where
$\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ and $p$ smaller than
the critical exponent $2_{N,k}^{\ast}:=\frac{2(N-k)}{N-k-2}$ for the Sobolev
embedding of $H^{1}(\mathbb{R}^{N-k})$ in $L^{q}(\mathbb{R}^{N-k})$, $1\leq
k\leq N-3.$ We show that in some suitable domains $\Omega$ there are positive
and sign changing solutions with positive and negative layers which concentrate
along one or several $k$-dimensional submanifolds of $\partial\Omega$ as $p$
approaches $2_{N,k}^{\ast}$ from below.
  Key words:Nonlinear elliptic boundary value problem; critical and
supercritical exponents; existence of positive and sign changing solutions.

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The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int