Minimax Theory and its Applications 07 (2022), No. 2, 339--364
Copyright 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.
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