Multiplicity of critical orbits to nonlinear, strongly indefinite functionals with sign-changing nonlinear part

We present an abstract critical point theorem about the existence of infinitely many critical orbits to strongly indefinite functionals with a sign-changing nonlinear part defined on a dislocation space with a discrete group action. We apply the abstract result to a Schrödinger equation aligned -Δ u + V(x) u = f(u) - λ g(u) aligned - Δ u + V ( x ) u = f ( u ) - λ g ( u ) with 0 in the spectral gap of the Schrödinger operator -Δ + V(x) - Δ + V ( x ) , that appears in nonlinear optics. We also consider equations with singular potentials arising from the study of cylindrically symmetric, electromagnetic waves to the system of Maxwell equations.