Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium

Druet, Olivier; Hebey, Emmanuel

doi:10.2140/apde.2009.2.305

Cited by 42 publications

(38 citation statements)

References 26 publications

(20 reference statements)

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“…There holds that D α ≤D α and by the analysis in Druet and Hebey [8], since (4.2) holds true, we can write that…”

Section: Proof Ofmentioning

confidence: 94%

“…The following key estimate is established in Druet and Hebey [8] (see also Druet, Hebey and Robert [10]). A slight difference here is that we need to handle the noncoercive case where ω = 0 and v = 0.…”

Section: Proof Ofmentioning

confidence: 98%

“…The proof we present is a shortcut with respect to the analysis in Druet and Hebey [9]. We mix in our analysis ideas from Li and Zhang [19], Druet and Hebey [8], Hebey and Robert [16], and Hebey, Robert and Wen [17]. The proof extends almost as it is to compact conformally flat manifolds of positive scalar curvature.…”

Section: Stability In the Critical Casementioning

confidence: 99%

“…We briefly sketch the proof and refer to Druet and Hebey [8] for more details. Given δ > 0 we define…”

Section: Proof Ofmentioning

confidence: 99%

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Schrödinger–Poisson systems in the 3-sphere

Hebey

Wei²

2012

Calc. Var.

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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:

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“…There holds that D α ≤D α and by the analysis in Druet and Hebey [8], since (4.2) holds true, we can write that…”

Section: Proof Ofmentioning

confidence: 94%

Section: Proof Ofmentioning

confidence: 98%

Section: Stability In the Critical Casementioning

confidence: 99%

“…We briefly sketch the proof and refer to Druet and Hebey [8] for more details. Given δ > 0 we define…”

Section: Proof Ofmentioning

confidence: 99%

See 2 more Smart Citations

Schrödinger–Poisson systems in the 3-sphere

Hebey

Wei²

2012

Calc. Var.

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show abstract

“…The condition turns out to be sharp. Assuming that (0.2) is an equality, then, see Druet and Hebey [16,18], we can construct various examples of unstable systems like (0.1) in any dimension n 6. These include the existence of clusters (multi peaks solutions with fewer geometrical blow-up points) and the existence of sequences (U α ) α of solutions with unbounded energy (namely such that U α H 1 → +∞ as α → +∞).…”

mentioning

confidence: 99%