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

Abstract

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) , the full Klein-Gordon-Maxwell-Proca system is written aswhere Δ g = −div g ∇ is the Laplace-Beltrami operator, Δ g = δd is half the Laplacian acting on forms, and δ is the codifferential. In its electrostatic form we assume A and ϕ do not depend on the time variable. Looking for standing waves solutions ψ(x, t) = u(x)e iωt , letting ϕ = ωv, there necessarily holds that A = 0 and the system reduces to the two following equations:In the above, m 0 , m 1 > 0 are masses (m 0 is the mass of the particle, m 1 is the Proca mass), and q > 0 is the electric charge of the particle. The Proca formalism comes with the assumption m 1 > 0. We refer to Section 1 for a discussion on the physics origin of the system. The system (0.2), in Proca form in closed manifolds, has been investigated in Druet and Hebey [5] and Hebey and Truong [8]. Existence of variational solutions and a priori bounds, which guarantee phase stability, were established in these papers. The existence of resonant states was left open. We answer the question in this paper.

Sign-changing solutions to elliptic second order equations: glueing a peak to a degenerate critical manifold

“…Here, ω k is the volume of the canonical unit k−sphere in R k+1 and K n is the best constant of the Sobolev inequality u 2 ≤ K ∇u 2 in R n . Finally, expanding J ε (u(ϕ)) with respect to ε and collecting (15), (16) and (17) yield…”

“…where (B ε ) ε is a bubble as defined in (6) (1): see for instance Rey [23] for a historical reference, Brendle-Marques [4] for the Yamabe equation, Druet-Hebey [13] and Esposito-Pistoia-Vétois [14] for perturbations of the Yamabe equation, Chen-Wei-Yan [6] and Hebey-Wei [15] for equations on the sphere, and the references therein. Sign-changing blowing-up solutions to (1) on the canonical sphere have been constructed by del Pino-Musso-Pacard-Pistoia [10,11] and Pistoia-Vétois [22].…”

Semiclassical states for a static supercritical Klein–Gordon–Maxwell–Proca system on a closed Riemannian manifold

“…So for the system on a closed manifold the Proca formalism is more interesting and more appropriate. We refer to [11] for a detailed discussion on KGMP-systems and their physical meaning.…”

Low energy solutions for singularly perturbed coupled nonlinear systems on a Riemannian manifold with boundary

“…The study of KGMP systems recently has known a rise of interest in the mathematical community. In [13,14,15] equation (1) has been studied on a Riemaniann boundariless manifold M . A similar problem has been considered in a flat domain Ω by D'Aprile and Wei [5,6].…”