Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$

Alberto Farina

doi:10.1016/j.crma.2005.07.006

Partial Differential Equations

Liouville-type results for solutions of

- Δ u = {| u |}^{p - 1} u

on unbounded domains of

R^{N}

Presented by: Philippe G. Ciarlet

Alberto Farina ¹

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, Volume 341 (2005) no. 7, pp. 415-418.

Abstract
Résumé

In this Note we study solutions, possibly unbounded and sign-changing, of the equation $- Δ u = {| u |}^{p - 1} u$ on unbounded domains of $R^{N}$ with $N ⩾ 2$ and $p > 1$ . We prove some Liouville-type results and a classification theorem for $C^{2}$ 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 $- Δ u = {| u |}^{p - 1} u$ dans des domaines non-bornés de $R^{N}$ 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.

Received: 2005-06-27
Accepted: 2005-06-29
Published online: 2005-09-19

DOI: 10.1016/j.crma.2005.07.006

Author's affiliations:

Alberto Farina 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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     author = {Alberto Farina},
     title = {Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {415--418},
     publisher = {Elsevier},
     volume = {341},
     number = {7},
     year = {2005},
     doi = {10.1016/j.crma.2005.07.006},
     language = {en},
}

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Alberto Farina. Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$. Comptes Rendus. Mathématique, Volume 341 (2005) no. 7, pp. 415-418. doi : 10.1016/j.crma.2005.07.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.006/

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