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

Riccardo Molle; Donato Passaseo

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]

Riccardo Molle ¹ ; Donato Passaseo 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, Volume 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 :

Riccardo Molle ¹ ; Donato Passaseo 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

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Riccardo Molle; Donato Passaseo. Positive solutions for slightly super-critical elliptic equations in contractible domains. Comptes Rendus. Mathématique, Volume 335 (2002) no. 5, pp. 459-462. doi : 10.1016/S1631-073X(02)02502-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02502-5/

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Riccardo Molle; Donato Passaseo Nonexistence results for elliptic problems with supercritical growth in thin planar domains, NoDEA. Nonlinear Differential Equations and Applications, Volume 30 (2023) no. 5, p. 23 (Id/No 66) | DOI:10.1007/s00030-023-00875-7 | Zbl:1522.35257
Riccardo Molle; Donato Passaseo Nonexistence of solutions for Dirichlet problems with supercritical growth in tubular domains, Advanced Nonlinear Studies, Volume 21 (2021) no. 1, pp. 189-198 | DOI:10.1515/ans-2021-2116 | Zbl:1487.35203
Riccardo Molle; Donato Passaseo Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth, Topological Methods in Nonlinear Analysis, Volume 57 (2021) no. 2, pp. 535-546 | Zbl:1479.35429
Riccardo Molle; Donato Passaseo Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni, Volume 31 (2020) no. 1, pp. 121-130 | DOI:10.4171/rlm/882 | Zbl:1437.35311
Teresa D'Aprile Multi-bubble solutions for a slightly supercritical elliptic problem in a domain with a small hole, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 105 (2016) no. 4, pp. 558-602 | DOI:10.1016/j.matpur.2015.11.008 | Zbl:1337.35034
Juanjuan Gao; Yong Zhang; Peihao Zhao Existence of positive solutions for a class of semilinear and quasilinear elliptic equations with supercritical case, Journal of Mathematical Analysis and Applications, Volume 381 (2011) no. 1, pp. 215-228 | DOI:10.1016/j.jmaa.2011.04.003 | Zbl:1220.35059
Angela Pistoia; Tobias Weth Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 24 (2007) no. 2, pp. 325-340 | DOI:10.1016/j.anihpc.2006.03.002 | Zbl:1166.35018
Riccardo Molle; Donato Passaseo Concentration phenomena for solutions of superlinear elliptic problems, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 23 (2006) no. 1, pp. 63-84 | DOI:10.1016/j.anihpc.2005.02.002 | Zbl:1293.35114
Riccardo Molle; Donato Passaseo Multiple solutions of supercritical elliptic problems in perturbed domains, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 23 (2006) no. 3, pp. 389-405 | DOI:10.1016/j.anihpc.2005.05.003 | Zbl:1172.35020
Riccardo Molle; Donato Passaseo Nonlinear elliptic equations with large supercritical exponents, Calculus of Variations and Partial Differential Equations, Volume 26 (2006) no. 2, pp. 201-225 | DOI:10.1007/s00526-005-0364-3 | Zbl:1093.35022
Riccardo Molle; Donato Passaseo Positive solutions of slightly supercritical elliptic equations in symmetric domains, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 21 (2004) no. 5, pp. 639-656 | DOI:10.1016/j.anihpc.2003.09.004 | Zbl:1149.35353

Cité par 11 documents. Sources : zbMATH

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