  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 is critical in
$\mathbb{R}^3$, i.e., $h(x,\mathbf{U})=|\mathbf{U}|^4\mathbf{U}$, or has
subcritical growth at infinity. If $h$ is radial in $\mathbf{U}$ 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 \[
-\Delta u+\frac{a}{r^2}u=f(x,u),\qquad u\colon\mathbb{R}^3\to \mathbb{R}, \]
where $x=(y,z)\in \mathbb{R}^2\times \mathbb{R}$, $r=|y|$ and $a \ge 0$. 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 subcritical case, instead, studying the Schr\"odinger
equation in higher dimensions, we find infinitely many bound states for both
problems.
