Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45)

Druet, Olivier; Hebey, Emmanuel; Robert, Frédéric

doi:10.1515/9781400826162

Cited by 109 publications

(173 citation statements)

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“…for all x ∈ S 3 and all α, and by standard properties of the Green's functions, see Druet, Hebey and Robert [10], by Lemma 4.4, and since u ∞ ≡ 0, we get that…”

Section: Proof Ofmentioning

confidence: 74%

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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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“…for all x ∈ S 3 and all α, and by standard properties of the Green's functions, see Druet, Hebey and Robert [10], by Lemma 4.4, and since u ∞ ≡ 0, we get that…”

Section: Proof Ofmentioning

confidence: 74%

“…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: 99%

Schrödinger–Poisson systems in the 3-sphere

Hebey

Wei²

2012

Calc. Var.

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“…Pioneering work also include Han [35] and Hebey and Vaugon [36] in the case of a Riemannian manifold. For s = γ = 0, the general pointwise estimates are performed in the monograph [21] of Druet-HebeyRobert. We also refer to Ghoussoub and Robert [29] for the optimal control with arbitrary high energy when s > 0 and γ = 0.…”

Section: Of the Bubble (B )mentioning

confidence: 99%

“…It consists of performing a more refined blow-up analysis on the minimizing sequences considered above. The proof-due to Ghoussoub and Robert [28]-uses the machinery developed in Druet et al [21] for equations of Yamabe-type on manifolds. It also allows to tackle problems with arbitrary high energy and not just minima [29].…”

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confidence: 99%

Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

Ghoussoub

Robert

2015

Bull. Math. Sci.

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In this survey paper, we consider variational problems involving the HardySchrödinger operator L γ := − − γ |x| 2 on a smooth domain of R n with 0 ∈ , and illustrate how the location of the singularity 0, be it in the interior of or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter can be stated as:where γ < n 2 4 , s ∈ [0, 2) and 2 (s) := 2(n−s) n−2 . We address questions regarding the explicit values of the optimal constant C := μ γ,s ( ), as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties often lead to situations where the best constants μ γ,s ( ) do not depend on the domain, and hence they are not attainable. We consider two different approaches for "breaking the homogeneity" of the problem, and restoring compactness. One approach was initiated by Brezis-Nirenberg, when γ = 0 and s = 0, and by Janelli, when γ > 0 and s = 0. It is suitable for the case where the singularity 0 is in the interior of , and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub-Kang for γ = 0, s > 0, and by C.S. Lin et al. and Ghoussoub-Robert, when γ = 0, s ≥ 0. It consists of considering domains, where the singularity 0 is on the boundary. Both of these approaches are rich in structure and in challenging problems. If 0 ∈ , then a negative linear perturbation suffices for higher dimensions, while a positive "Hardy-singular interior mass" theorem for the operator L γ is required in lower dimensions. If the singularity 0 belongs to the boundary ∂ , then the local geometry around 0 (convexity and mean curvature) plays a crucial role in high dimensions, while a positive "Hardy-singular boundary mass" theorem is needed for the lower dimensions. Each case leads to a distinct notion of critical dimension for the operator L γ .

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“…They also turn out to be key points in the analysis of ruling out bubbling and proving compactness of solutions. Possible references in book form on these subjects are Druet-Hebey-Robert [26], Ghoussoub [36], and Struwe [63].…”

Section: Introductionmentioning

confidence: 99%