Comptes Rendus
no. 13-14
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

Christopher Grumiau1 ; Christophe Troestler1
1 Institut de mathématique, Université de Mons-Hainaut, place du parc 20, B-7000 Mons, Belgium
Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 767-771.

Soit l'équation −Δu=λ2|u|p−2u avec conditions au bord de Dirichlet, où Ω⊆R2 est ouvert borné, λ2 la deuxième valeur propre de −Δ et p>2. Nous prouvons que, sur un convexe de classe C2, 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 −Δu=λ2|u|p−2u with Dirichlet boundary conditions, where the domain Ω is an open bounded subset of R2, λ2 is the second eigenvalue of −Δ, and p>2. We prove that, if Ω is C2 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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.04.023
Christopher Grumiau 1 ; Christophe Troestler 1

1 Institut de mathématique, Université de Mons-Hainaut, place du parc 20, B-7000 Mons, Belgium 
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Christopher Grumiau; Christophe Troestler. Nodal line structure of least energy nodal solutions for Lane–Emden problems. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 767-771. doi : 10.1016/j.crma.2009.04.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.023/

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