Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiplicity and concentration of solutions for a fractional Kirchhoff
  equation with magnetic field and critical growth   We investigate the existence, multiplicity and concentration of nontrivial
solutions for the following fractional magnetic Kirchhoff equation with
critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3}
[u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u
\quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon$ is a
small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4},
1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent,
$(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian,
$A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential,
$V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous potential
verifying the global condition due to Rabinowitz \cite{Rab}, and
$f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^{1}$ subcritical nonlinearity. Due
to the presence of the magnetic field and the critical growth of the
nonlinearity, several difficulties arise in the study of our problem and a
careful analysis will be needed. The main results presented here are
established by using minimax methods, concentration compactness principle of
Lions \cite{Lions}, a fractional Kato's type inequality and the
Ljusternik-Schnirelmann theory of critical points.

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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: econ