Nonlinear Curl-Curl Problems in R3
Keywords:
Time-harmonic Maxwell equations, ground state, variational methods, strongly indefi nite functional, curl-curl problem, Orlicz spaces, N-functions.Abstract
We survey recent results concerning ground states and bound states u: R3 → R3 to the curl-curl problem ∇×(∇×u)+V(x)u=f(x,u)
in R3, which originates from the nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of ∇×(∇×·). 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.
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