On the prescribed <i>Q</i>-curvature problem on $ {S}^{n}$

Hichem Chtioui; Afef Rigane

doi:10.1016/j.crma.2010.03.018

Partial Differential Equations/Differential Geometry

On the prescribed Q-curvature problem on

S^{n}

[Sur le problème de la Q-courbure prescrite sur

S^{n}

]

Présenté par : Pierre-Louis Lions

Hichem Chtioui ¹ ; Afef Rigane ¹

¹ Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia

Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 635-638.

Résumé
Abstract

Dans cette Note nous prescrivons une courbure du quatrième order-la Q-courbure sur la sphère standard de dimension $n ⩾ 5$ . Sous une « condition de platitude » d'ordre $β \in [n - 4, n [$ au voisinage de chaque point critique de la fonction Q-courbure prescrite, nous prouvons un nouveau résultat d'existence grâce à une formule de type Euler–Hopf. Notre argument donne une minoration du nombre des métriques ayant la même Q-courbure.

In this Note we prescribe a fourth order curvature – the Q-curvature on the standard n-sphere, $n ⩾ 5$ . Under the “flatness condition” of order β, $n - 4 ⩽ β < n$ near each critical point of the prescribed Q-curvature function, we prove new existence result through an Euler–Hopf type formula. Our argument gives a lower bound on the number of conformal metrics having the same Q-curvature.

Reçu le : 2010-03-01
Accepté le : 2010-03-29
Publié le : 2010-04-29

DOI : 10.1016/j.crma.2010.03.018

Affiliations des auteurs :

Hichem Chtioui ¹ ; Afef Rigane ¹

¹ Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia

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     title = {On the prescribed {\protect\emph{Q}-curvature} problem on $ {S}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {635--638},
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     volume = {348},
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     year = {2010},
     doi = {10.1016/j.crma.2010.03.018},
     language = {en},
}

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Hichem Chtioui; Afef Rigane. On the prescribed Q-curvature problem on $ {S}^{n}$. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 635-638. doi : 10.1016/j.crma.2010.03.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.018/

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