@Article{Mederski2019,
author="Mederski, Jaros{\l}aw
and Schino, Jacopo
and Szulkin, Andrzej",
title="Multiple Solutions to a Nonlinear Curl--Curl Problem in {\$}{\$}{\{}{\backslash}mathbb {\{}R{\}}{\}}^3{\$}{\$}R3",
journal="Archive for Rational Mechanics and Analysis",
year="2019",
month="Nov",
day="21",
abstract="We look for ground states and bound states {\$}{\$}E:{\{}{\backslash}mathbb {\{}R{\}}{\}}^3{\backslash}rightarrow {\{}{\backslash}mathbb {\{}R{\}}{\}}^3{\$}{\$}E:R3{\textrightarrow}R3 to the curl--curl problem {\$}{\$}{\backslash}begin{\{}aligned{\}} {\backslash}nabla {\backslash}times ({\backslash}nabla {\backslash}times E)= f(x,E) {\backslash}qquad {\backslash}text {\{} in {\}} {\{}{\backslash}mathbb {\{}R{\}}{\}}^3, {\backslash}end{\{}aligned{\}}{\$}{\$}∇{\texttimes}(∇{\texttimes}E)=f(x,E)inR3,which originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of {\$}{\$}{\backslash}nabla {\backslash}times ({\backslash}nabla {\backslash}times {\backslash}cdot ){\$}{\$}∇{\texttimes}(∇{\texttimes}{\textperiodcentered}). The growth of the nonlinearity f is controlled by an N-function {\$}{\$}{\backslash}Phi :{\{}{\backslash}mathbb {\{}R{\}}{\}}{\backslash}rightarrow [0,{\backslash}infty ){\$}{\$}$\Phi$:R{\textrightarrow}[0,∞) such that {\$}{\$}{\backslash}displaystyle {\backslash}lim {\_}{\{}s{\backslash}rightarrow 0{\}}{\backslash}Phi (s)/s^6={\backslash}lim {\_}{\{}s{\backslash}rightarrow +{\backslash}infty {\}}{\backslash}Phi (s)/s^6=0{\$}{\$}lims{\textrightarrow}0$\Phi$(s)/s6=lims{\textrightarrow}+∞$\Phi$(s)/s6=0. We prove the existence of a ground state, that is, 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 for our problem have not been studied so far in {\$}{\$}{\{}{\backslash}mathbb {\{}R{\}}{\}}^3{\$}{\$}R3 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 Schr{\"o}dinger equations.",
issn="1432-0673",
doi="10.1007/s00205-019-01469-3",
url="https://doi.org/10.1007/s00205-019-01469-3"
}