Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniqueness and nondegeneracy of solutions for a critical nonlocal
  equation   The aim of this paper is to classify the positive solutions of the nonlocal
critical equation: $$ -\Delta u=\left(I_{\mu}\ast
u^{2^{\ast}_{\mu}}\right)u^{{2}^{\ast}_{\mu}-1},~~x\in\mathbb{R}^{N}, $$ where
$0<\mu<N$, if $N=3\ \hbox{or} \ 4$ and $0<\mu\leq4$ if $N\geq5$, $I_{\mu}$ is
the Riesz potential defined by $$
I_{\mu}(x)=\frac{\Gamma(\frac{\mu}{2})}{\Gamma(\frac{N-\mu}{2})\pi^{\frac{N}{2}}2^{N-\mu}|x|^{\mu}}
$$ with $\Gamma(s)=\int^{+\infty}_{0}x^{s-1}e^{-x}dx$, $s>0$ and
$2^{\ast}_{\mu}=\frac{2N-\mu}{N-2}$ is the upper critical exponent in the sense
of the Hardy-Littlewood-Sobolev inequality. We apply the moving plane method in
integral forms to prove the symmetry and uniqueness of the positive solutions
and prove the nondegeneracy of the unique solutions for the equation when $\mu$
close to $N$.

26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
19
The categories of this paper are: eess
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The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat