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Comptes Rendus Mathématique
Volume 339, n° 5
pages 339-344 (septembre 2004)
Doi : 10.1016/j.crma.2004.07.004
accepted : 4 July 2004
Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains
Propriétés qualitatives des solutions nodales pour des problèmes elliptiques semilinéaires dans des domaines à symétrie sphérique.
 

Amandine Aftalion a , Filomena Pacella b
a Laboratoire Jacques-Louis Lions, B.C.187, université Paris 6, 175, rue du Chevaleret, 75013 Paris, France 
b Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro 2, 00185 Roma, Italy 

Abstract

We study the qualitative properties of sign changing solutions of the Dirichlet problem   in ,   on , where is a ball or an annulus and f is a   function with  . We prove that any radial sign changing solution has a Morse index bigger or equal to   and give sufficient conditions for the nodal surface of a solution to intersect the boundary. In particular, we prove that any least energy nodal solution is non radial and its nodal surface touches the boundary. To cite this article: A. Aftalion, F. Pacella, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

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Résumé

Nous étudions les propriétés qualitatives des solutions qui changent de signe du problème de Dirichlet   dans ,   sur , où est une boule ou un anneau et f une fonction   avec  . Nous prouvons que toute solution radiale qui change de signe a un indice de Morse supérieur ou égal à   et donnons des conditions suffisantes pour que la surface nodale intersecte le bord. En particulier, nous prouvons que toute solution nodale d'énergie minimale est non radiale et sa surface nodale touche le bord. Pour citer cet article : A. Aftalion, F. Pacella, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.


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