Nonlinear elliptic equations with critical Sobolev exponent in nearly starshaped domains
[Équations elliptiques non linéaire avec non-linéarité critique en ouverts presque étoilés]
Comptes Rendus. Mathématique, Tome 335 (2002) no. 12, pp. 1029-1032.

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 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.

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Accepté le :
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DOI : 10.1016/S1631-073X(02)02614-6
Molle, Riccardo 1 ; Passaseo, Donato 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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Molle, Riccardo; Passaseo, Donato. Nonlinear elliptic equations with critical Sobolev exponent in nearly starshaped domains. Comptes Rendus. Mathématique, Tome 335 (2002) no. 12, pp. 1029-1032. doi : 10.1016/S1631-073X(02)02614-6. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02614-6/

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