High frequency periodic solutions of semilinear equations

Allain, Geneviève; Beaulieu, Anne

doi:10.1016/j.crma.2007.07.010

Partial Differential Equations

High frequency periodic solutions of semilinear equations
[Solutions périodiques de haute fréquence d'équations semi-linéaires]

Présenté par : Haïm Brezis

Allain, Geneviève ¹ ; Beaulieu, Anne ²

¹ Laboratoire d'analyse et de mathématiques appliquées, Université Paris-Est, UMR CNRS 8050, Faculté de sciences et technologie, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France
² Laboratoire d'analyse et de mathématiques appliquées, Université Paris-Est, UMR CNRS 8050, 5, boulevard Descartes, 77454 Marne-la-Vallée cedex 2, France

Comptes Rendus. Mathématique, Tome 345 (2007) no. 7, pp. 381-384.

Résumé
Abstract

On s'intéresse aux solutions positives de $- ε^{2} Δ u + f (u) = 0$ dans $S^{1} \times R$ , c'est-à-dire aux solutions périodiques en $x_{1}$ , la première coordonnée. Le cas modèle est $f (u) = u - u^{p}$ , $p > 1$ . Nous prouvons que, pour ε suffisamment grand, toute solution positive est une fonction de $x_{2}$ seulement.

We are interested with positive solutions of $- ε^{2} Δ u + f (u) = 0$ in $S^{1} \times R$ , i.e. periodic solutions in the first coordinate $x_{1}$ . The model function for f is $f (u) = u - u^{p}$ , $p > 1$ . We prove that for ε large enough, any positive solution is a function of the second coordinate only.

Reçu le : 2007-03-18
Accepté le : 2007-07-18
Publié le : 2007-09-20

DOI : 10.1016/j.crma.2007.07.010

Affiliations des auteurs :

Allain, Geneviève ¹ ; Beaulieu, Anne ²

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     title = {High frequency periodic solutions of semilinear equations},
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     pages = {381--384},
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Allain, Geneviève; Beaulieu, Anne. High frequency periodic solutions of semilinear equations. Comptes Rendus. Mathématique, Tome 345 (2007) no. 7, pp. 381-384. doi : 10.1016/j.crma.2007.07.010. http://www.numdam.org/articles/10.1016/j.crma.2007.07.010/

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