The effect of curvature on the best constant in the Hardy–Sobolev inequalities

Ghoussoub, Nassif; Robert, Frédéric

doi:10.1007/s00039-006-0579-2

Cited by 99 publications

(113 citation statements)

References 27 publications

(51 reference statements)

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“…• The other approach was initiated by Ghoussoub and Kang [25] and developed by Ghoussoub and Robert [28][29][30] when s > 0 and γ = 0, and by C.S. Lin et al [37,[45][46][47] and Ghoussoub and Robert [31] when γ = 0.…”

Section: + |X| (2−s)β + (γ )mentioning

confidence: 99%

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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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Section: + |X| (2−s)β + (γ )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].…”

mentioning

confidence: 99%

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

Ghoussoub

Robert

2015

Bull. Math. Sci.

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“…Actually, by the results in [10], extremals for Q(s, Ω) exists if Ω is "average concave in a neighborhood of the origin". Later on, in the same year, Ghoussoub and Robert [15,16] used refined blow-up analysis to prove existence of an extremal for Q(s, Ω) provided the mean curvature of ∂Ω is negative at 0.…”

Section: Introductionmentioning

confidence: 99%

The role of the mean curvature in a Hardy–Sobolev trace inequality

Fall¹,

Minlend²,

Thiam³

2015

Nonlinear Differ. Equ. Appl.

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The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.

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“…The existence results of (1.3) were proved in [5] by the global compactness method. Moreover, Ghoussonb and Robert in [6] have proved that µ 2 * (s),s (Ω) is achieved if 0 ∈ ∂Ω. In [9], Hsai et al use the blow-up method to prove that the following elliptic equation involving two critical exponents…”

Section: Introductionmentioning

confidence: 99%

Existence of critical elliptic systems with boundary singularities

Jiang¹,

Zhou²

2013

OpMath

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Abstract. In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponentwhere N ≥ 4 and Ω is a C 1 bounded domain in R N with 0 ∈ ∂Ω. 0 < s < 2, α + β = 2, α, β > 1, λ > 0 and 1 < p <. The case when 0 belongs to the boundary of Ω is closely related to the mean curvature at the origin on the boundary. We show in this paper that problem ( * ) possesses at least a positive solution.

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The effect of curvature on the best constant in the Hardy–Sobolev inequalities

Cited by 99 publications

References 27 publications

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

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

The role of the mean curvature in a Hardy–Sobolev trace inequality

Existence of critical elliptic systems with boundary singularities

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