Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ground States of a Nonlinear Curl-Curl Problem in Cylindrically
  Symmetric Media   We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U +
V(x) U= \Gamma(x)|U|^{p-1}U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell
equations for monochromatic fields. We search for solutions as minimizers
(ground states) of the corresponding energy functional defined on subspaces
(defocusing case) or natural constraints (focusing case) of
$H(\mathrm{curl};\mathbb{R}^3)$. Under a cylindrical symmetry assumption on the
functions $V$ and $\Gamma$ the variational problem can be posed in a symmetric
subspace of $H(\mathrm{curl};\mathbb{R}^3)$. For a strongly defocusing case
$\mathrm{esssup}\, \Gamma <0$ with large negative values of $\Gamma$ at
infinity we obtain ground states by the direct minimization method. For the
focusing case $\mathrm{essinf}\, \Gamma >0$ the concentration compactness
principle produces ground states under the assumption that zero lies outside
the spectrum of the linear operator $\nabla \times \nabla \times +V(x)$.
Examples of cylindrically symmetric functions $V$ are provided for which this
holds.

15
The categories of this paper are: nucl-ex
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
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The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph