Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiple solutions to cylindrically symmetric curl-curl problems and
  related Schr\"odinger equations with singular potentials   We look for multiple solutions $\mathbf{U}\colon\mathbb{R}^3\to\mathbb{R}^3$
to the curl-curl problem \[
\nabla\times\nabla\times\mathbf{U}=h(x,\mathbf{U}),\qquad x\in\mathbb{R}^3, \]
with a nonlinear function
$h\colon\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}^3$ which has subcritical
growth at infinity or is critical in $\mathbb{R}^3$, i.e.
$h(x,\mathbf{U})=|\mathbf{U}|^4\mathbf{U}$. If $h$ is radial in $\mathbf{U}$,
$N=3$, $K=2$ and $a=1$ below, then we show that the solutions to the problem
above are in one to one correspondence with the solutions to the following
Schr\"odinger equation \[ -\Delta u+\frac{a}{r^2}u=f(x,u),\qquad
u\colon\mathbb{R}^N\to \mathbb{R}, \] where $x=(y,z)\in \mathbb{R}^K\times
\mathbb{R}^{N-K}$, $N>K\geq 2$, $r=|y|$ and $a>a_0\in(-\infty,0]$. In the
critical case, the multiplicity problem for the latter equation has been
studied only in the autonomous case $a=0$ and the available methods seem to be
insufficient for the problem involving the singular potential, i.e. $a\neq 0$,
due to the lack of conformal invariance. Therefore we develop methods for the
critical curl-curl problem and show the multiplicity of bound states for both
equations in the case $N=3$, $K=2$ and $a=1$. In the subcritical case, instead,
applying a critical point theory to the Schr\"odinger equation above, we find
infinitely many bound states for both problems.

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The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
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The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci