Nodal line structure of least energy nodal solutions for Lane–Emden problems

Grumiau, Christopher; Troestler, Christophe

doi:10.1016/j.crma.2009.04.023

Partial Differential Equations

Nodal line structure of least energy nodal solutions for Lane–Emden problems
[Structure de la ligne nodale des solutions nodales d'énergie minimale pour le problème de Lane–Emden]

Présenté par : Haïm Brezis

Grumiau, Christopher ¹ ; Troestler, Christophe ¹

¹ Institut de mathématique, Université de Mons-Hainaut, place du parc 20, B-7000 Mons, Belgium

Comptes Rendus. Mathématique, Tome 347 (2009) no. 13-14, pp. 767-771.

Résumé
Abstract

Soit l'équation $- Δ u = λ_{2} {| u |}^{p - 2} u$ avec conditions au bord de Dirichlet, où $Ω \subseteq R^{2}$ est ouvert borné, $λ_{2}$ la deuxième valeur propre de −Δ et $p > 2$ . Nous prouvons que, sur un convexe de classe $C^{2}$ , la ligne nodale de toute solution nodale d'énergie minimale intersecte ∂Ω pour p proche de 2. Par ailleurs, nous montrons également l'existence d'un ensemble connexe mais non simplement connexe, tel que, pour p proche de 2, la ligne nodale de toute solution nodale d'énergie minimale n'intersecte pas ∂Ω.

In this Note, we consider the Lane–Emden problem $- Δ u = λ_{2} {| u |}^{p - 2} u$ with Dirichlet boundary conditions, where the domain Ω is an open bounded subset of $R^{2}$ , $λ_{2}$ is the second eigenvalue of −Δ, and $p > 2$ . We prove that, if Ω is $C^{2}$ and convex, the nodal line intersects ∂Ω when p is close to 2. In contrast, we also exhibit a connected — but not simply connected — domain Ω such that, for p close to 2, the nodal line does not intersect ∂Ω.

Reçu le : 2008-12-04
Accepté le : 2009-04-20
Publié le : 2009-05-20

DOI : 10.1016/j.crma.2009.04.023

Affiliations des auteurs :

Grumiau, Christopher ¹ ; Troestler, Christophe ¹

¹ Institut de mathématique, Université de Mons-Hainaut, place du parc 20, B-7000 Mons, Belgium

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     title = {Nodal line structure of least energy nodal solutions for {Lane{\textendash}Emden} problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {767--771},
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     doi = {10.1016/j.crma.2009.04.023},
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Grumiau, Christopher; Troestler, Christophe. Nodal line structure of least energy nodal solutions for Lane–Emden problems. Comptes Rendus. Mathématique, Tome 347 (2009) no. 13-14, pp. 767-771. doi : 10.1016/j.crma.2009.04.023. http://archive.numdam.org/articles/10.1016/j.crma.2009.04.023/

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