Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

Caffarelli, Luis A.; Gidas, Basilis; Spruck, Joel

doi:10.1002/cpa.3160420304

Cited by 1,100 publications

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“…wherev > 0 satisfies 2 2m 2 0 ∆v =v 5 and is given by the Caffarelli-Gidas-Spruck [5] classification. Given λ > 0 and x ∈ R 3 , we let…”

Section: Stability In the Critical Casementioning

confidence: 99%

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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“…wherev > 0 satisfies 2 2m 2 0 ∆v =v 5 and is given by the Caffarelli-Gidas-Spruck [5] classification. Given λ > 0 and x ∈ R 3 , we let…”

Section: Stability In the Critical Casementioning

confidence: 99%

Schrödinger–Poisson systems in the 3-sphere

Hebey

Wei²

2012

Calc. Var.

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“…The fundamental work on (0.4) is due to Fowler [6,7]. Other references include [5,20,27,30], to cite just a few. In Propositions 3.4, 3.5, and 3.7 we recall or prove some results about (0.4) (and the generalized version of (0.4)).…”

Section: This Equation Is Called the Lane-emden Equation It Arises Imentioning

confidence: 99%

On the Cauchy Problem for Reaction-Diffusion Equations

Wang¹

1993

Transactions of the American Mathematical Society

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Abstract. The simplest model of the Cauchy problem considered in this paper is the following . . u, =Au + uP, x£Rn,t>0, u>0, p>\, w u\l=0 = (t> € CB(R"), 4>>0, ?0.It is well known that when 1 < p < (n + 2)/n, the local solution of (*) blows up in finite time as long as the initial value 4> is nontrivial; and when p > (n + 2)/n , if is "small", (*) has a global classical solution decaying to zero as t -» +oo, while if for global existence and to study the asymptotic behavior of those global solutions.In particular, we prove that if n > 3 , p > n/(n -2),(us is a singular equilibrium of (*)) where 0 < À < 1 , then (*) has a (unique) global classical solution u with 0 < u < Xus and u{x, t)<((Xx-p -l)(p-l)0"1/(p_l).(This result implies that uq = 0 is stable w.r.t. to a weighted L°° topology when n > 3 and p > n/(n -2).) We also obtain some sufficient conditions on for global nonexistence and those conditions, when combined with our global existence result, indicate that for (¡> around us, we are in a delicate situation, and when p is fixed, uq = 0 is "increasingly stable" as the dimension n Î +oo . A slightly more general version of (*) is also considered and similar results are obtained. IntroductionThe simplest model of the Cauchy problem considered in this paper is the following:u, = Au + up, x £ R", t > 0, p > 1, u\t=0 = >0, tf>£0.(0.1 ) is related to many equations arising from mathematical biology and chemical reactor theory, and the results for (0.1) may be used to the study of those equations as shown by Aronson and Weinberger [1]. Besides the practical interest in it, (0.1) and its various generalizations have been model problems for

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“…In the semilinear case p = 2 it has been proved in the celebrated paper [2] (see also [3]) that any solution to −∆u = u N+2 N−2 (N ≥ 3) is radial and hence classified by (1.1). It is crucial in the proof the use of the Kelvin transform that allows to reduce to the study of the symmetry of solutions that have nice decaying properties at infinity.…”

Section: Introductionmentioning

confidence: 99%

Classification of positive D1,p(RN)-solutions to the critical p

Sciunzi

2016

Advances in Mathematics

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Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth

Cited by 1,100 publications

References 7 publications

Schrödinger–Poisson systems in the 3-sphere

Schrödinger–Poisson systems in the 3-sphere

On the Cauchy Problem for Reaction-Diffusion Equations

Classification of positive D1,p(RN)-solutions to the critical p

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