Nonlinear elliptic equations with critical Sobolev exponent in nearly starshaped domains

Riccardo Molle; Donato Passaseo

doi:10.1016/S1631-073X(02)02614-6

Nonlinear elliptic equations with critical Sobolev exponent in nearly starshaped domains
[Équations elliptiques non linéaire avec non-linéarité critique en ouverts presque étoilés]

Riccardo Molle ¹ ; Donato Passaseo ²

¹ Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Roma, Italy
² Dipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy

Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1029-1032.

Résumé
Abstract

On montre que, si $Ω$ satisfait certaines conditions, le problème (1) ci-dessous, pour ε>0 suffisamment petit et k grand, admet des solutions qui pour ε→0 se concentrent et explosent exactement en k points ; les points de concentration s'approchent du bord de $Ω$ quand k→∞ ; le nombre de solutions est arbitrairement grand pourvu que ε soit suffisamment petit. Parmi les ouverts bornés $Ω$ qui satisfont ces conditions il y en a aussi de contractibles, qui peuvent même être arbitrairement proches de ouverts étoilés.

Under suitable assumptions on $Ω$ , we show that, for ε>0 small and k large enough, problem (1) below has solutions which concentrate and blow-up as ε→0 at exactly k points; the blowing-up points approach $\partial Ω$ as k→∞; the number of solutions tends to infinity as ε→0. These assumptions allow $Ω$ to be contractible and even arbitrarily close to starshaped domains.

Reçu le : 2002-10-01
Accepté le : 2002-10-25
Publié le : 2002-11-21

DOI : 10.1016/S1631-073X(02)02614-6

Affiliations des auteurs :

Riccardo Molle ¹ ; Donato Passaseo ²

¹ Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Roma, Italy
² Dipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy

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     author = {Riccardo Molle and Donato Passaseo},
     title = {Nonlinear elliptic equations with critical {Sobolev} exponent in nearly starshaped domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1029--1032},
     publisher = {Elsevier},
     volume = {335},
     number = {12},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02614-6},
     language = {en},
}

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Riccardo Molle; Donato Passaseo. Nonlinear elliptic equations with critical Sobolev exponent in nearly starshaped domains. Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1029-1032. doi : 10.1016/S1631-073X(02)02614-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02614-6/

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