Comptes Rendus
Partial Differential Equations
Elliptic equations with critical exponent on S3: new non-minimising solutions
Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 391-394.

Consider the problem:

{ΔS3U=λU+U5,U>0on B,U=0on B,
where B is a ball on S3 with geodesic radius θ1, and ΔS3 is the Laplace–Beltrami operator on S3. We prove that for any θ1(π/2,π) and any k>1, there exist at least 2k solutions of this problem for λ sufficiently large negative.

On considère le problème :

{ΔS3U=λU+U5,U>0sur B,U=0sur B,
B est une boule sur S3 de rayon geodésique θ1, et ΔS3 est l'opérateur Laplace–Beltrami sur S3. On montre que pour tout θ1(π/2,π), et tout k>1, ce problème possède au moins 2k solutions pour λ<0 avec |λ| assez grand.

Accepted:
Published online:
DOI: 10.1016/j.crma.2004.07.010
Haïm Brezis 1; Lambertus A. Peletier 2

1 Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, BC 187, 4, place Jussieu, 75252 Paris cedex 05, France
2 Mathematical Institute, Leiden University, PB 9512, 2300 RA Leiden, The Netherlands
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Haïm Brezis; Lambertus A. Peletier. Elliptic equations with critical exponent on $ {\mathbf{S}}^{3}$: new non-minimising solutions. Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 391-394. doi : 10.1016/j.crma.2004.07.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.010/

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[2] A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part II, Indiana Univ. Math. J., in press

[3] C. Bandle; R. Benguria The Brezis–Nirenberg problem on S3, J. Differential Equations, Volume 178 (2002), pp. 264-279

[4] C. Bandle; L.A. Peletier Best constants and Emden equations for the critical exponent in S3, Math. Ann., Volume 313 (1999), pp. 83-93

[5] H. Brezis; L. Nirenberg Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., Volume 36 (1983), pp. 437-477

[6] L.A. Peletier; W.C. Troy Spatial Patterns: Higher Order Models in Physics and Mechanics, Birkhäuser, Boston, 2001

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[8] S.I. Stingelin, Das Brezis–Nirenberg-Problem auf der Sphäre Sn, Inauguraldissertation, Univerität Basel, 2004

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