Multiple solutions to cylindrically symmetric curl-curl problems and related Schr\"odinger equations with singular potentials

Abstract

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.