Resonant states for the static Klein–Gordon–Maxwell–Proca system

Abstract

We prove the existence of resonant states for the critical static Klein-Gordon-Maxwell-Proca system in the case of closed manifolds. Standing waves solutions with arbitarilly large multi-spikes amplitudes and unstable phases are constructed. We investigate in this paper the existence of resonant states for the electrostatic Klein-Gordon-Maxwell-Proca system in closed manifolds, a massive version of the more traditional electrostatic Klein-Gordon-Maxwell system. The system provides a dualistic model for the description of the interaction between a charged relativistic matter scalar field and the electromagnetic field that it generates. The external vector field (ϕ, A) in the system inherits a mass and is governed by the Proca action which generalizes that of Maxwell. Let (M, g) be a closed three-dimensional Riemannian manifold. Writing the matter scalar field in polar form as ψ(x, t )= u(x, t)e iS(x,t) ,t he