We find solutions E : Ω → R 3 of the problemin Ω ν × E = 0 on ∂Ω on a simply connected, smooth, bounded domain Ω ⊂ R 3 with connected boundary and exterior normal ν : ∂Ω → R 3 . Here ∇× denotes the curl operator in R 3 , the nonlinearity F : Ω × R 3 → R is superquadratic and subcritical in E. The model nonlinearity is of the form F (x, E) = Γ(x)|E| p for Γ ∈ L ∞ (Ω) positive, some 2 < p < 6. It need not be radial nor even in the E-variable. The problem comes from the time-harmonic Maxwell equations, the boundary conditions are those for Ω surrounded by a perfect conductor.MSC 2010: Primary: 35Q60; Secondary: 35J20, 58E05, 78A25
We investigate the existence of solutions E : R 3 → R 3 of the time-harmonic semilinear Maxwell equation0 almost everywhere on R 3 , ∇× denotes the curl operator in R 3 and F : R 3 × R 3 → R is a nonlinear function in E. In particular we find a ground state solution provided that suitable growth conditions on F are imposed and the L 3/2 -norm of V is less than the best Sobolev constant. In applications, F is responsible for the nonlinear polarization and V (x) = −μω 2 ε(x) where μ > 0 is the magnetic permeability, ω is the frequency of the time-harmonic electric field {E(x)e iωt } and ε is the linear part of the permittivity in an inhomogeneous medium.
We look for ground states and bound states E : R 3 → R 3 to the curl-curl problemwhich 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 :We prove the existence of a ground state, i.e. 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 R 3 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. 2 2010 Mathematics Subject Classification. Primary: 35Q60; Secondary: 35J20, 78A25.In general J ′ is not (sequentially) weak-to-weak * continuous, however we show the weakto-weak * continuity of J ′ for sequences on the topological manifold M. Obviously, the same regularity holds for E ′ and M E .for any (φ, ψ) ∈ V × W.
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