
Minimax Theory and its Applications 07 (2022), No. 2, 339364 Copyright Heldermann Verlag 2022 Nonlinear CurlCurl Problems in R^{3} Jaroslaw Mederski Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland jmederski@impan.pl Jacopo Schino Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland jschino@impan.pl [Abstractpdf] We survey recent results concerning ground states and bound states $u\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curlcurl problem $$ \nabla\times(\nabla\times u)+V(x)u= f(x,u) \qquad\hbox{ in } \mathbb{R}^3, $$ which originates from the 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 superlinear and subcritical at infinity or purely critical and we demonstrate a variational approach to the problem involving the generalized Nehari manifold. We also present some refinements of known results. Keywords: Timeharmonic Maxwell equations, ground state, variational methods, strongly indefinite functional, curlcurl problem, Orlicz spaces, Nfunctions. MSC: 35Q60; 35J20, 78A25. [ Fulltextpdf (201 KB)] for subscribers only. 