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

Gaczkowski, Michał; Mederski, Jarosław; Schino, Jacopo

## Abstract:

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. ... mehr

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht/Preprint Publikationsjahr 2020 Sprache Englisch Identifikator ISSN: 2365-662X KITopen-ID: 1000120066 Verlag Karlsruher Institut für Technologie (KIT) Umfang 21 S. Serie CRC 1173 Preprint ; 2020/18 Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019) Schlagwörter Maxwell equations, curl-curl problem, quintic effect, Schrödinger equation, singular potential, variational methods
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