Comptes Rendus
Partial Differential Equations
On the Brezis–Nirenberg problem on S3, and a conjecture of Bandle–Benguria
[Sur l'équation de Brezis–Nirenberg sur S3 et une conjecture de Bandle–Benguria]
Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 153-156.

We consider the following Brezis–Nirenberg problem on S3

ΔS3u=λu+u5inD,u>0inDandu=0on D,
where D is a geodesic ball on S3 with geodesic radius θ1, and ΔS3 is the Laplace–Beltrami operator on S3. We prove that for any λ<34 and for every θ1<π with πθ1 sufficiently small (depending on λ), there exists bubbling solution to the above problem. This solves a conjecture raised by Bandle and Benguria [J. Differential Equations 178 (2002) 264–279] and Brezis and Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394].

Nous considérons le problème de Brezis–Nirenberg suivant sur S3

ΔS3u=λu+u5dansD,u>0dansDetu=0surD,
D est une boule géodésique sur S3 de rayon géodésique θ1, et ΔS3 est l'opérateur de Laplace–Beltrami sur S3. Nous montrons que pour tout λ<34 et tout θ1<π avec πθ1 suffisamment petit (dependant de λ), il existe des solutions pour le problème précédent. Ce résultat répond à une conjecture de Bandle et Benguria [J. Differential Equations 178 (2002) 264–279] et de Brezis et Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291–394].

Reçu le :
Publié le :
DOI : 10.1016/j.crma.2005.06.001

Wenyi Chen 1 ; Juncheng Wei 2

1 Department of Mathematics, Wuhan University, Wuhan, Hubei 430072, PR China
2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
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Wenyi Chen; Juncheng Wei. On the Brezis–Nirenberg problem on $ {\mathbf{S}}^{3}$, and a conjecture of Bandle–Benguria. Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 153-156. doi : 10.1016/j.crma.2005.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.001/

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