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

Brezis, Haïm; Peletier, Lambertus A.

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

Brezis, Haïm ¹ ; Peletier, Lambertus A.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, Tome 339 (2004) no. 6, pp. 391-394

Abstract (VO)
Résumé

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.

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.

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

DOI : 10.1016/j.crma.2004.07.010

Affiliations des auteurs :

Brezis, Haïm ¹ ; Peletier, Lambertus A. 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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     author = {Brezis, Ha{\"\i}m and Peletier, Lambertus A.},
     title = {Elliptic equations with critical exponent on $ {\mathbf{S}}^{3}$: new non-minimising solutions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {391--394},
     year = {2004},
     publisher = {Elsevier},
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     number = {6},
     doi = {10.1016/j.crma.2004.07.010},
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Brezis, Haïm; Peletier, Lambertus A. Elliptic equations with critical exponent on $ {\mathbf{S}}^{3}$: new non-minimising solutions. Comptes Rendus. Mathématique, Tome 339 (2004) no. 6, pp. 391-394. doi: 10.1016/j.crma.2004.07.010

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