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

Mederski, Jarosław; Schino, Jacopo; Szulkin, Andrzej

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.

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DOI: 10.5445/IR/1000089035
Veröffentlicht am 04.01.2019
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht
Jahr 2019
Sprache Englisch
Identifikator ISSN: 2365-662X
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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