Elliptic equations with critical growth and a large set of boundary singularities

Ghoussoub, Nassif; Robert, Frédéric

doi:10.1090/s0002-9947-09-04655-8

Cited by 18 publications

(21 citation statements)

References 27 publications

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“…In this situation, we suspect that the sign if the mean curvature of ∂Ω at a point might influence on the the existence of minimizers. Finally we note that Ghoussoub and Robert in [14] obtained several results for the case Γ a subspace of dimension k ≥ 2, and among other results, if Γ intersects ∂Ω transversely, they obtain existence results under some negativity assumptions on the mean curvature.…”

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confidence: 69%

“…The first paper, to our knowledge, being the one of Ghoussoub and Kang [17] who considered the Hardy-Sobolev inequality with singularity at the boundary. For more results in this direction, see the works of Ghoussoub and Robert in [12][13][14]16], Demyanov and Nazarov [8], Chern and Lin [7], Lin and Li [19], the authors and Minlend in [11] and the references there in. We point out that in the pure Hardy-Sobolev case, σ ∈ (0, 2), with singularity at the boundary, one has existence of minimizers for every dimension N ≥ 3 as long as the mean curvature of the boundary is negative at the point singularity, see [15].…”

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confidence: 99%

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Hardy-Sobolev inequality with singularity a curve

Fall¹,

Thiam²

2017

TMNA

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Abstract. We consider a bounded domain Ω of R N , N ≥ 3, and h a continuous function on Ω. Let Γ be a closed curve contained in Ω. We study existence of positive solutions u ∈ H 1 0 (Ω) to the equation, σ ∈ (0, 2), and ρ Γ is the distance function to Γ. For N ≥ 4, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case N = 3, we obtain existence of ground-state solution provided the trace of the regular part of the Green of −∆ + h is positive at a point of the curve.

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confidence: 69%

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confidence: 99%

Hardy-Sobolev inequality with singularity a curve

Fall¹,

Thiam²

2017

TMNA

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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%

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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“…If D R N , D 0, t D 1, the positive extremals have been completely identified in [20,21], and the non-degeneracy of the positive extremals has been proved in [22]. For more results, we can refer to [23][24][25][26][27][28][29][30]. However, for the case t ¤ 1, it seems that there are no results on the existence of solutions for problem (1.1) because in this case, the explicit form or the asymptotic properties of the positive extremals of (1.1) with D 0 in R N are unknown, which makes it impossible to verify that the Palais-Smale sequences corresponding to the functional I are precompact by the standard argument in [6].…”

Section: Introductionmentioning

confidence: 99%

Infinitely many solutions for a Hardy–Sobolev equation involving critical growth

Peng

Wang

2013

Math Methods in App Sciences

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In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev equation with critical growth:Á 1 q and positive constants (possibly different) by C. B D B r .x 0 / denotes an open ball with radius r and centerx 0 , and we write B for the open ball with radius r and center x 0 , 8x 0 2 . The organization of the paper is as follows. In Section 2, we will give some integral estimates. In Section 3, we obtain some estimates on safe regions. We will prove our main result in Section 4. In order that we can give a clear line of our framework, we will list some estimates for linear problems with Hardy-Sobolev potential, an iteration result, and the decomposition of approximating solutions in Appendices A, B, and C.

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Elliptic equations with critical growth and a large set of boundary singularities

Cited by 18 publications

References 27 publications

Hardy-Sobolev inequality with singularity a curve

Hardy-Sobolev inequality with singularity a curve

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

Infinitely many solutions for a Hardy–Sobolev equation involving critical growth

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