Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotic behavior of least energy solutions for a fractional Laplacian
  eigenvalue problem on $R^N$   We are interested in the existence and asymptotical behavior for the least
energy solutions of the following fractional eigenvalue problem
  \begin{equation*}
  (P)\quad (-\Delta)^{s}u+V(x)u=\mu u+am(x)|u|^{\frac{4s}{N}}u,\quad
\int_{\mathbb{R}^{N}}|u|^{2}dx=1,\ u\in H^{s}(\mathbb{R}^{N}),
  \end{equation*}
  where $s\in(0,1)$, $\mu\in\mathbb{R}$, $a>0$, $V(x)$ and $m(x)$ are
$L^{\infty}(\mathbb{R}^{N})$ functions with $N\geq2$. We prove that there is a
threshold $a_s^{*}>0$ such that problem $(P)$ has a least energy solution
$u_{a}(x)$ for each $a\in(0,a_s^{*})$ and $u_{a}$ blows up, as $a\nearrow
a_s^{*}$, at some point $x_0 \in \mathbb{R}^N$, which makes $V(x_0)$ be the
minimum and $m(x_0)$ be the maximum. Moreover, the precise blowup rates for
$u_a$ are obtained under suitable conditions on $V(x)$ and $m(x)$.

10
The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex