Comptes Rendus
Partial Differential Equations/Differential Geometry
On the prescribed scalar curvature problem on Sn: The degree zero case
[Sur le problème de courbure scalaire prescrite sur Sn : Le cas de degré zéro]
Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 583-586.

Dans cette Note nous considérons le problème dʼexistence de métriques conformes avec courbure scalaire prescrite, sur la sphère standard Sn, n3. Nous donnons de nouveaux résultats dʼexistence et de multiplicité reposant sur un nouveau type de formule dʼEuler–Hopf. Nos arguments ont également lʼavantage dʼétendre des résultats bien connus de Y. Li (1995) [10].

In this Note, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere Sn, n3. We give new existence and multiplicity results based on a new Euler–Hopf formula type. Our argument also has the advantage of extending the well known results due to Y. Li (1995) [10].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.06.012

Randa Ben Mahmoud 1 ; Hichem Chtioui 2 ; Afef Rigane 1

1 Department of Mathematics, Faculty of Sciences of Sfax, Route of Soukra, Sfax, Tunisia
2 Department of Mathematics, King Abdulaziz University, P.O. 80230, Jeddah, Saudi Arabia
@article{CRMATH_2012__350_11-12_583_0,
     author = {Randa Ben Mahmoud and Hichem Chtioui and Afef Rigane},
     title = {On the prescribed scalar curvature problem on $ {S}^{n}$: {The} degree zero case},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {583--586},
     publisher = {Elsevier},
     volume = {350},
     number = {11-12},
     year = {2012},
     doi = {10.1016/j.crma.2012.06.012},
     language = {en},
}
TY  - JOUR
AU  - Randa Ben Mahmoud
AU  - Hichem Chtioui
AU  - Afef Rigane
TI  - On the prescribed scalar curvature problem on $ {S}^{n}$: The degree zero case
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 583
EP  - 586
VL  - 350
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2012.06.012
LA  - en
ID  - CRMATH_2012__350_11-12_583_0
ER  - 
%0 Journal Article
%A Randa Ben Mahmoud
%A Hichem Chtioui
%A Afef Rigane
%T On the prescribed scalar curvature problem on $ {S}^{n}$: The degree zero case
%J Comptes Rendus. Mathématique
%D 2012
%P 583-586
%V 350
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2012.06.012
%G en
%F CRMATH_2012__350_11-12_583_0
Randa Ben Mahmoud; Hichem Chtioui; Afef Rigane. On the prescribed scalar curvature problem on $ {S}^{n}$: The degree zero case. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 583-586. doi : 10.1016/j.crma.2012.06.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.06.012/

[1] A. Ambrosetti; M. Badiale Homoclinics: Poincaré–Melnikov type results via variational approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 15 (1998), pp. 233-252

[2] A. Ambrosetti; J. Garcia Azorero; A. Peral Perturbation of Δu+u(N+2)(N2)=0, the scalar curvature problem in RN and related topics, J. Funct. Anal., Volume 165 (1999), pp. 117-149

[3] A. Bahri Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math., vol. 182, Longman Sci. Tech., Harlow, 1989

[4] A. Bahri An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimensions, Duke Math. J., Volume 81 (1996), pp. 323-466

[5] R. Ben Mahmoud; H. Chtioui Existence results for the prescribed scalar curvature on S3, Ann. Inst. Fourier, Volume 61 (2011), pp. 971-986

[6] R. Ben Mahmoud; H. Chtioui Prescribing the scalar curvature problem on higher-dimensional manifolds, Discrete Contin. Dyn. Syst. A, Volume 32 (2012) no. 5, pp. 1857-1879

[7] R. Ben Mahmoud, H. Chtioui, A. Rigane, On the prescribed scalar curvature on Sn: The degree zero case, in preparation.

[8] A. Chang; P. Yang A perturbation result in prescribing scalar curvature on Sn, Duke Math. J., Volume 64 (1991), pp. 27-69

[9] J. Kazdan; J. Warner Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature, Ann. Math., Volume 101 (1975), pp. 317-331

[10] Y.Y. Li Prescribing scalar curvature on Sn and related topics, Part I, J. Diff. Eq., Volume 120 (1995), pp. 319-410

[11] Y.Y. Li Prescribing scalar curvature on Sn and related topics, Part II: Existence and compactness, Comm. Pure Appl. Math., Volume 49 (1996), pp. 541-579

Cité par Sources :

Commentaires - Politique