Multiple solutions to cylindrically symmetric curl-curl problems and related Schrödinger equations with singular potentials

We look for multiple solutions $\mathbf{U}:\mathbb{R}^3\to\mathbb{R}^3$ 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 $h:\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}^3$ 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 Schrödinger equation $$-\Delta u+\frac{a}{r^2}u=f(x,u),\qquad u:\mathbb{R}^N\to\mathbb{R},$$ where $x=(y, z)\in\mathbb{R}^K\times\mathbb{R}^{N-K}$, $N>K\ge2$, $r=|y|$ and $a>a_0\in(−∞, 0]$. In the subcritical case, applying a critical point theory to the Schrödinger equation above, we find infinitely many bound states for both problems. In the critical case, however, 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. $a\ne0$, 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 $a=1$