Regularity of the extremal solutions for the Liouville system - Archive ouverte HAL Accéder directement au contenu
Communication Dans Un Congrès Année : 2013

Regularity of the extremal solutions for the Liouville system

Résumé

In this short note, we study the smoothness of the extremal solutions to the following system of equations: (1)      −∆u = µe v in Ω, −∆v = λe u in Ω, u = v = 0 on ∂Ω, where λ, µ > 0 are parameters and Ω is a smoothly bounded domain of R N , N ≥ 1. As shown by M. Montenegro (see [6]), there exists a limiting curve Υ in the first quadrant of the (λ, µ)-plane serving as borderline for existence of classical solutions of (1). He also proved the existence of a weak solution u * for every (λ * , µ *) on the curve Υ and left open the question of its regularity. Following standard terminology (see e.g. the books [3], [5] for an introduction to this vast subject), u * is called an extremal solution. Our result is the following. Theorem 1 Let 1 ≤ N ≤ 9. Then, extremal solutions to (1) are smooth.
Fichier principal
Vignette du fichier
DFS_corrected.pdf (150.34 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01111689 , version 1 (30-01-2015)

Identifiants

Citer

L Dupaigne, Alberto Farina, B Sirakov. Regularity of the extremal solutions for the Liouville system. Geometric Partial Differential Equations and Nonlinear Analysis, Oct 2012, Pisa, Italy, Italy. pp.139 - 144, ⟨10.1007/978-88-7642-473-1_7⟩. ⟨hal-01111689⟩
74 Consultations
63 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More