Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonlinear time-harmonic Maxwell equations in domains   The search for time-harmonic solutions of nonlinear Maxwell equations in the
absence of charges and currents leads to the elliptic equation
$$\nabla\times\left(\mu(x)^{-1} \nabla\times u\right) - \omega^2\varepsilon(x)u
= f(x,u)$$ for the field $u:\Omega\to\mathbb{R}^3$ in a domain
$\Omega\subset\mathbb{R}^3$. Here $\varepsilon(x) \in \mathbb{R}^{3\times3}$ is
the (linear) permittivity tensor of the material, and $\mu(x) \in
\mathbb{R}^{3\times3}$ denotes the magnetic permeability tensor. The
nonlinearity $f:\Omega\times\mathbb{R}^3\to\mathbb{R}^3$ comes from the
nonlinear polarization. If $f=\nabla_uF$ is a gradient then this equation has a
variational structure. The goal of this paper is to give an introduction to the
problem and the variational approach, and to survey recent results on ground
and bound state solutions. It also contains refinements of known results and
some new results.

21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin