Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains

Amandine Aftalion; Filomena Pacella

doi:10.1016/j.crma.2004.07.004

Partial Differential Equations

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.]

Présenté par : Haïm Brezis

Amandine Aftalion ¹ ; Filomena Pacella ²

¹ Laboratoire Jacques-Louis Lions, B.C.187, université Paris 6, 175, rue du Chevaleret, 75013 Paris, France
² Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro 2, 00185 Roma, Italy

Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 339-344.

Résumé
Abstract

Nous étudions les propriétés qualitatives des solutions qui changent de signe du problème de Dirichlet $Δ u + f (u) = 0$ dans Ω, $u = 0$ sur ∂Ω, où Ω est une boule ou un anneau et f une fonction $C^{1}$ avec $f (0) ⩾ 0$ . Nous prouvons que toute solution radiale qui change de signe a un indice de Morse supérieur ou égal à $N + 1$ 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.

We study the qualitative properties of sign changing solutions of the Dirichlet problem $Δ u + f (u) = 0$ in Ω, $u = 0$ on ∂Ω, where Ω is a ball or an annulus and f is a $C^{1}$ function with $f (0) ⩾ 0$ . We prove that any radial sign changing solution has a Morse index bigger or equal to $N + 1$ 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.

Accepté le : 2004-07-04
Publié le : 2004-08-10

DOI : 10.1016/j.crma.2004.07.004

Affiliations des auteurs :

Amandine Aftalion ¹ ; Filomena Pacella ²

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     author = {Amandine Aftalion and Filomena Pacella},
     title = {Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {339--344},
     publisher = {Elsevier},
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     number = {5},
     year = {2004},
     doi = {10.1016/j.crma.2004.07.004},
     language = {en},
}

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Amandine Aftalion; Filomena Pacella. Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 339-344. doi : 10.1016/j.crma.2004.07.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.004/

Bibliographie
Cité par

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[2] T. Bartsch, T. Weth, M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math. (2004), in press

[3] H. Brezis; L. Nirenberg Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., Volume 36 (1983) no. 4, pp. 437-477

[4] A. Castro; J. Cossio; J. Neuberger A sign-changing solution for a superlinear Dirichlet problem, Rocky Mt. J. Math., Volume 27 (1997) no. 4, pp. 1041-1053

[5] G. Chen; J. Zhou; W.M. Ni Algorithms and visualization for solutions of nonlinear elliptic equations, Int. J. Bifur. Chaos Appl. Sci. Eng., Volume 10 (2000) no. 7, pp. 1565-1612

[6] J. Wei; M. Winter Symmetry of nodal solutions for singularly perturbed elliptic problems on a ball, Indiana Univ. Math. J. (2004)

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