High frequency periodic solutions of semilinear equations

Geneviève Allain; Anne Beaulieu

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

Geneviève Allain ¹ ; Anne Beaulieu 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, Volume 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 :

Geneviève Allain ¹ ; Anne Beaulieu ²

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     author = {Genevi\`eve Allain and Anne Beaulieu},
     title = {High frequency periodic solutions of semilinear equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {381--384},
     publisher = {Elsevier},
     volume = {345},
     number = {7},
     year = {2007},
     doi = {10.1016/j.crma.2007.07.010},
     language = {en},
}

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PY  - 2007
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EP  - 384
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%A Anne Beaulieu
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Geneviève Allain; Anne Beaulieu. High frequency periodic solutions of semilinear equations. Comptes Rendus. Mathématique, Volume 345 (2007) no. 7, pp. 381-384. doi : 10.1016/j.crma.2007.07.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.07.010/

Bibliographie
Cité par

[1] G. Alberti; L. Ambrosio; X. Cabré On a long-standing conjecture of E. De Giorgi: old and recent results, Acta Appl. Math., Volume 65 (2001), pp. 9-33

[2] L. Almeida; L. Damascelli; Y. Ge A few symmetry results for nonlinear elliptic PDE on noncompact manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 19 (2002) no. 3, pp. 313-342

[3] H. Berestycki; F. Hamel; R. Monneau One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., Volume 103 (2000) no. 3, pp. 375-396

[4] H. Berestycki; L. Nirenberg On the method of moving plane and the sliding method, Bol. Soc. Bras. Mat., Volume 22 (1991), pp. 1-39

[5] E.N. Dancer New solutions of equations on $R^{n}$ , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume XXX (2001), pp. 535-563

[6] N. Ghoussoub; C. Gui On a conjecture of De Giorgi and some related problems, Math. Ann., Volume 311 (1998), pp. 481-491

[7] B. Gidas; W.M. Ni; L. Nirenberg Symmetry of positive solutions of nonlinear elliptic equations in $R^{n}$ , Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies, vol. 7A, Academic Press, New York, 1981, pp. 369-402

[8] D. Gilbarg; N. Trudinger Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983

[9] M.K. Kwong; L. Zhang Uniqueness of the positive solution of $Δ u + f (u) = 0$ in an annulus, Differential Integral Equations, Volume 4 (1991), pp. 583-599

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