Elliptic equations with critical exponent on $ {\mathbf{S}}^{3}$: new non-minimising solutions

Haïm Brezis; Lambertus A. Peletier

doi:10.1016/j.crma.2004.07.010

Partial Differential Equations

Elliptic equations with critical exponent on

S^{3}

: new non-minimising solutions
[Équations elliptiques avec exposant critique sur

S^{3}

: nouvelles solutions non-minimisantes.]

Présenté par : Haïm Brezis

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

Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 391-394.

Résumé
Abstract

On considère le problème :

{\begin{matrix} - Δ_{S^{3}} U = λ U + U^{5}, U > 0 & sur B^{'}, \\ U = 0 & sur \partial B^{'}, \end{matrix}

où

B^{'}

est une boule sur

S^{3}

de rayon geodésique

θ_{1}

, et

Δ_{S^{3}}

est l'opérateur Laplace–Beltrami sur

S^{3}

. On montre que pour tout

θ_{1} \in (π / 2, π)

, et tout

k > 1

, ce problème possède au moins 2k solutions pour

λ < 0

avec

| λ |

assez grand.

Consider the problem:

{\begin{matrix} - Δ_{S^{3}} U = λ U + U^{5}, U > 0 & on B^{'}, \\ U = 0 & on \partial B^{'}, \end{matrix}

where

B^{'}

is a ball on

S^{3}

with geodesic radius

θ_{1}

, and

Δ_{S^{3}}

is the Laplace–Beltrami operator on

S^{3}

. We prove that for any

θ_{1} \in (π / 2, π)

and any

k > 1

, there exist at least 2k solutions of this problem for λ sufficiently large negative.

Accepté le : 2004-07-05
Publié le : 2004-08-27

DOI : 10.1016/j.crma.2004.07.010

Affiliations des auteurs :

Haïm Brezis ¹ ; Lambertus A. Peletier ²

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     author = {Ha{\"\i}m Brezis and Lambertus A. Peletier},
     title = {Elliptic equations with critical exponent on $ {\mathbf{S}}^{3}$: new non-minimising solutions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {391--394},
     publisher = {Elsevier},
     volume = {339},
     number = {6},
     year = {2004},
     doi = {10.1016/j.crma.2004.07.010},
     language = {en},
}

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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/

Bibliographie
Cité par

[1] A. Ambrosetti; A. Malchiodi; W.-M. Ni Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Commun. Math. Phys., Volume 235 (2003), pp. 427-466

[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 $S^{3}$ , 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 $S^{3}$ , 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

[7] S.I. Pohozaev; S.I. Pohozaev Eigenfunctions of the equation $Δ u + λ f (u) = 0$ , Soviet Math., Volume 165 (1965), pp. 36-39 (in Russian)

[8] S.I. Stingelin, Das Brezis–Nirenberg-Problem auf der Sphäre $S^{n}$ , Inauguraldissertation, Univerität Basel, 2004

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