Geodesic
Partial Differential Equations
Liouville-type results for solutions of −Δu=|u|p−1u on unbounded domains of RN
[Résultats de type Liouville pour des solutions de −Δu=|u|p−1u dans des domaines non-bornés de RN]

Présenté par : Philippe G. Ciarlet

Farina, Alberto1
1 LAMFA, CNRS UMR 6140, université de Picardie Jules Verne, faculté de mathématiques et d'informatique, 33, rue Saint-Leu, 80039 Amiens, France
Comptes Rendus. Mathématique, Tome 341 (2005) no. 7, pp. 415-418.

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In this Note we study solutions, possibly unbounded and sign-changing, of the equation −Δu=|u|p−1u on unbounded domains of RN with N⩾2 and p>1. We prove some Liouville-type results and a classification theorem for C2 solutions belonging to one of the following classes: stable solutions, finite Morse index solutions and solutions which are stable outside a compact set. We also extend, to smooth coercive epigraphs, the well-known results of Gidas and Spruck concerning non-negative solutions of the considered equation.

Cette Note porte sur l'étude des solutions, éventuellement non-bornées et de signe quelconque, de l'équation −Δu=|u|p−1u dans des domaines non-bornés de RN avec N⩾2 et p>1. Nous démontrons des résultats de type Liouville ainsi que des théorèmes de classification pour les solutions régulières appartenant à une des classes suivantes : solutions stables, solutions d'indice de Morse fini et solutions stables à l'extérieur d'un compact. Nous étendons aussi, au cas d'un épigraphe coercif régulier, les célèbres résultats de Gidas et Spruck concernant les solutions positives de l'équation considérée.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2005.07.006

Farina, Alberto 1

1 LAMFA, CNRS UMR 6140, université de Picardie Jules Verne, faculté de mathématiques et d'informatique, 33, rue Saint-Leu, 80039 Amiens, France 
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Farina, Alberto. Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$. Comptes Rendus. Mathématique, Tome 341 (2005) no. 7, pp. 415-418. doi : 10.1016/j.crma.2005.07.006. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2005.07.006/

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