Schrödinger–Poisson systems in the 3-sphere

Abstract

Abstract. We investigate nonlinear Schrödinger-Poisson systems in the 3-sphere. We prove existence results for these systems and discuss the question of the stability of the systems with respect to their phases. While, in the subcritical case, we prove that all phases are stable, we prove in the critical case that there exists a sharp explicit threshold below which all phases are stable and above which resonant frequencies and multi-spikes blowing-up solutions can be constructed. Solutions of the Schrödinger-Poisson systems are standing waves solutions of the electrostatic Maxwell-Schrödinger system. Stable phases imply the existence of a priori bounds on the amplitudes of standing waves solutions. Unstable phases give rise to resonant states.We investigate in this paper nonlinear Schrödinger-Poisson systems in the 3-sphere. These are electrostatic versions of the Maxwell-Schrödinger system which describes the evolution of a charged nonrelativistic quantum mechanical particle interacting with the electromagnetic field it generates. We adopt here the Proca formalism. Then the particle interacts via the minimum coupling rulewith an external massive vector field (ϕ, A) which is governed by the MaxwellProca Lagrangian. In particular, we recover as part of the full system the massive modified Maxwell equations in SI units, which are hereafter explicitly written down:These massive Maxwell equations, as modified to Proca form, appear to have been first written in modern format by Schrödinger [25]. The Proca formalism a priori breaks Gauge invariance. Gauge invariance can be restored by the Stueckelberg trick, as pointed out by Pauli [21], and then by the Higgs mechanism. We refer to Goldhaber and Nieto [14,15], Luo, Gillies and Tu [20], and Ruegg and Ruiz-Altaba [24] for very complete references on the Proca approach. In the electrostatic case of the Maxwell-Schrödinger system, looking for standing waves solutions, we are led to the nonlinear Schrödinger-Poisson system we investigate in this paper. It is stated as follows:

Non-resonant states for Schrödinger–Poisson critical systems in high dimensions

“…This system arises when looking for standing wave solutions of the massive version of the Maxwell equations modified according to the Proca formalism (see Hebey and Wei [10]). Closely related systems have been investigated by Alves and Souto [1], by Azzollini et al [2], Benci and Fortunato [4], Benci and Bonanno [3], Bonanno [5,6], Candela and Salvatore [7], Coclite and Holden [8], Ianni and Vaira [11], Pisani and Siciliano [12,13], Ruiz and Siciliano [14], and Zhang and Sun [17].…”

“…Closely related systems have been investigated by Alves and Souto [1], by Azzollini et al [2], Benci and Fortunato [4], Benci and Bonanno [3], Bonanno [5,6], Candela and Salvatore [7], Coclite and Holden [8], Ianni and Vaira [11], Pisani and Siciliano [12,13], Ruiz and Siciliano [14], and Zhang and Sun [17]. For closed manifolds, (1.1) has been investigated when n = 3 by Hebey and Wei [10], and when n = 4, 5 by Thizy [15,16]. In particular, blowing-up sequences of solutions of (1.1) have been proved to exist in the case of S 3 for particular values of ω in Hebey and Wei [10], and for all ω in the case of S 1 × N and T 2 × N , where N is an arbitrary closed 3-manifold in Thizy [15].…”

“…For closed manifolds, (1.1) has been investigated when n = 3 by Hebey and Wei [10], and when n = 4, 5 by Thizy [15,16]. In particular, blowing-up sequences of solutions of (1.1) have been proved to exist in the case of S 3 for particular values of ω in Hebey and Wei [10], and for all ω in the case of S 1 × N and T 2 × N , where N is an arbitrary closed 3-manifold in Thizy [15]. In this short note we prove that, surprisingly, blowing-up sequences do not exist anymore when n ≥ 6.…”

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

“…More precisely we use here the so-called localized energy method (see Del Pino et al [4], Rey and Wei [12], and Wei [17]) which goes through the choice of suitable approximate solutions and the use of finite-dimensional reduction. The proof we present here follows closely the lines of Hebey and Wei [9].…”

“…Following Hebey and Wei [9], Del Pino et al [4], and Rey and Wei [12], we obtain that there existε 0 > 0 and C > 0 such that for anyε ∈ (0,ε 0 ) and any …”