A priori bounds for positive solutions of nonlinear elliptic equations

Gidas, Basilis; Spruck, Joel

doi:10.1080/03605308108820196

Cited by 747 publications

(591 citation statements)

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“…Moreover, by (3.4), we have that 2 2m 2 0 ∆ũ =ũ p−1 , a contradiction with the Liouville result of Gidas and Spruck [13]. Hence, (3.3) cannot happen, and for any (ω α ) α such that ω α → ω as α → +∞, and any (u α , v α ) solutions of (4.1), there exists C > 0 such that u α L ∞ ≤ C. By the second equation in (3.1) it follows that u α L ∞ + v α L ∞ ≤ C for all α, and by standard elliptic theory, a C 2,θ -bound holds as well.…”

Section: Stability In the Subcritical Casementioning

confidence: 49%

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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Section: Stability In the Subcritical Casementioning

confidence: 49%

Schrödinger–Poisson systems in the 3-sphere

Hebey

Wei²

2012

Calc. Var.

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“…The proof is based on a scaling method of [28] and two non-existence results (see theorems 3.1 and 3.4).…”

Section: Proof Of Theorem 12 and Further Resultsmentioning

confidence: 99%

“…The proof of this last result relies on the classical argument of rescaling introduced in [28], which yields to problems on unbounded domains. Therefore, some Liouvilletype results are required, and this is the point where the restriction α 1 appears.…”

Section: (Iii) If λ > λ There Is No Solutionmentioning

confidence: 99%

A concave—convex elliptic problem involving the fractional Laplacian

Brändle¹,

Colorado²,

Pablo³

et al. 2013

Proceedings of the Royal Society of Edinburgh: Section A Mathem

502

421

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“…We call a pair (β 0 1 , β 0 2 ) a blow-up pair if it has the property to be in the intersection of two of those lines, and further β 0 is to the left of or on l 11 , below or on l 22 , below or on l 12 , and above or on l 21 . Suppose the a ij (x) corresponding to these two (or more) lines are positive on Ω.…”

Section: Then (Aps) Holdsmentioning

confidence: 99%

“…We will use a blow-up type argument, originally due to Gidas and Spruck [21] in the case of a scalar equation, and developed for our type of systems in [18] -we refer to that paper for details. Set …”

Section: Proposition 22mentioning

confidence: 99%