Multiple Solutions to a Nonlinear Curl-Curl Problem in R3

We look for ground states and bound states E :R³→R³ to the curl-curl problem ∇ ×(∇ ×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 ∇ ×(∇ ×.) . The growth of the nonlinearity f is controlled by an N-function Φ :R →[0 ,∞ ) such that lims →0 Φ (s ) /s⁶=lims →+∞ Φ (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 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 Schrödinger equations.