Positive solutions for slightly super-critical elliptic equations in contractible domains

Molle, Riccardo; Passaseo, Donato

doi:10.1016/S1631-073X(02)02502-5

Positive solutions for slightly super-critical elliptic equations in contractible domains
[Solutions positives pour l'équation Δu+u^{(n+2)/(n−2)+ε}=0 en ouverts contractibles]

Molle, Riccardo ¹ ; Passaseo, Donato ²

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

Comptes Rendus. Mathématique, Tome 335 (2002) no. 5, pp. 459-462.

Résumé
Abstract

On donne des exemples d'ouverts bornés $Ω$ , même contractibles, satisfaisant la propriété suivante : il existe $\bar{k} (Ω)$ tel que, pour tout $k ⩾ \bar{k} (Ω)$ , le problème $P (ϵ, Ω)$ ci-dessous, pour ε>0 suffisamment petit, a des solutions qui pour ε→0 explosent exactement en k points. On prouve aussi que ces points convergent vers des points de $\partial Ω$ quand k→∞.

We give examples of bounded domains $Ω$ , even contractible, having the following property: there exists $\bar{k} (Ω)$ such that, for every integer $k ⩾ \bar{k} (Ω)$ , problem $P (ϵ, Ω)$ below, for ε>0 small enough, has at least one solution blowing up as ε→0 at exactly k points. We also prove that the blow-up points tend to some points of $\partial Ω$ as k→∞.

Reçu le : 2002-07-08
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02502-5

Affiliations des auteurs :

Molle, Riccardo ¹ ; Passaseo, Donato ²

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Molle, Riccardo; Passaseo, Donato. Positive solutions for slightly super-critical elliptic equations in contractible domains. Comptes Rendus. Mathématique, Tome 335 (2002) no. 5, pp. 459-462. doi : 10.1016/S1631-073X(02)02502-5. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02502-5/

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