Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Minimax Theory and its Applications 07 (2022), No. 2, 339--364Copyright Heldermann Verlag 2022 Nonlinear Curl-Curl Problems in R3 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 [Abstract-pdf] We survey recent results concerning ground states and bound states $u\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl 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: Time-harmonic Maxwell equations, ground state, variational methods, strongly indefinite functional, curl-curl problem, Orlicz spaces, N-functions. MSC: 35Q60; 35J20, 78A25. [ Fulltext-pdf  (201  KB)] for subscribers only.