Multiple positive solutions for singularly perturbed elliptic problems in exterior domains

✍ Scribed by Giovanna Cerami; Riccardo Molle

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 149 KB
• Volume: 20
• Category: Article
• ISSN: 0294-1449
• DOI: 10.1016/s0294-1449(02)00030-6

✦ Synopsis

The equation -ε 2 u + a ε (x)u = u p-1 with boundary Dirichlet zero data is considered in an exterior domain = R N \ ω (ω bounded and N 2). Under the assumption that a ε a 0 > 0 concentrates round a point of as ε → 0, that p > 2 and p < 2N/(N -2) when N 3, the existence of at least three positive distinct solutions is proved.  2003 Éditions scientifiques et médicales Elsevier SAS MSC: 35J20; 35J60 Keywords: Exterior domains; Lack of compactness; Multiplicity of solutions RÉSUMÉ. -Dans cet article on étude l'équation -ε 2 u + a ε (x)u = u p-1 dans l'ouvert extérieur = R N \ ω (ω borné et N 2), avec la condition de Dirichlet u = 0 sur ∂ . En supposant que a ε a 0 > 0 se concentre autour d'un point du domaine quand ε → 0, que p > 2 et que p < 2N/(N -2) quand N 3, on démontre que le problème possède au moins trois solutions distinctes.  2003 Éditions scientifiques et médicales Elsevier SAS ✩ Work supported by Italian M.I.U.R., national research project "Metodi variazionali e topologici nello studio di fenomeni non lineari".