Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiplicity and concentration results for fractional
  Schr\"odinger-Poisson equations with magnetic fields and critical growth   We deal with the following fractional Schr\"odinger-Poisson equation with
magnetic field \begin{equation}
\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{2^{*}_{s}-2}u
\quad \mbox{in} \mathbb{R}^{3}, \nonumber \end{equation} where $\varepsilon>0$
is a small parameter, $s\in (\frac{3}{4}, 1)$, $t\in (0,1)$,
$2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent,
$(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian,
$V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential,
$A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential and
$f:\mathbb{R}\rightarrow \mathbb{R}$ is a subcritical nonlinearity. Under a
local condition on the potential $V$, we study multiplicity and concentration
of nontrivial solutions as $\varepsilon\rightarrow 0$. In particular, we relate
the number of nontrivial solutions with the topology of the set where the
potential $V$ attains its minimum.

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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: supr-con
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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: hep-th
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin