# Multiple solutions to a nonlinear curl-curl problem in $\mathbb{R}^3$ (Preliminary version)

Mederski, Jarosław; Schino, Jacopo; Szulkin, Andrzej

Abstract:
We look for ground states and bound states $E:\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem
$$\nabla\times(\nabla\times E)=f(x,E)\qquad\text{ in }\mathbb{R}^3$$
which originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla\times(\nabla\times{}\cdot{})$. The growth of the nonlinearity $f$ is controlled by an $N$-function $\Phi:\mathbb{R}\to[0,\infty)$ such that $\displaystyle\lim_{s\to0}\Phi(s)/s^6=\lim_{s\to+\infty}\Phi(s)/s^6=0$. We prove the existence of a ground state, i.e. a least energy nontrivial solution, and the existence of infinitely many geometrically distinct bound states. We improve previous results concerning ground states of curl-curl problems. Multiplicity results have not been studied so far in $\mathbb{R}^3$ and in order to do this we construct a suitable critical point theory. It is applicable to a wide class of strongly indefinite problems, including this one and Schrödinger equations.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2019 Sprache Englisch Identifikator ISSN: 2365-662X URN: urn:nbn:de:swb:90-890350 KITopen-ID: 1000089035 Auflage Preliminary version Verlag KIT, Karlsruhe Umfang 33 S. Serie CRC 1173 ; 2019/1 Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015) Schlagworte time-harmonic Maxwell equations, ground state, variational methods, strongly indefinite functional, curl-curl problem
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