Comptes Rendus
Partial Differential Equations
Nodal line structure of least energy nodal solutions for Lane–Emden problems
Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 767-771.

In this Note, we consider the Lane–Emden problem Δu=λ2|u|p2u 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.

Soit l'équation Δu=λ2|u|p2u avec conditions au bord de Dirichlet, où ΩR2 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.

Received:
Accepted:
Published online:
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/

[1] A. Aftalion; F. Pacella Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains, C. R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 5, pp. 339-344 MR MR2092460 (2005f:35086)

[2] G. Alessandrini Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains, Comment. Math. Helv., Volume 69 (1994) no. 1, pp. 142-154 MR MR1259610 (95d:35111)

[3] A. Ambrosetti; P.H. Rabinowitz Dual variational methods in critical point theory and applications, J. Funct. Anal., Volume 14 (1973), pp. 349-381 MR MR0370183 (51 #6412)

[4] T. Bartsch; T. Weth; M. Willem Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., Volume 96 (2005), pp. 1-18 (MR MR2177179)

[5] D. Bonheure; V. Bouchez; C. Grumiau; J. Van Schaftingen Asymptotics and symmetries of least energy nodal solutions of Lane–Emden problems with slow growth, Comm. Contemporary Math., Volume 10 (2008) no. 4, pp. 609-631 (MR MR2444849)

[6] H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983, Théorie et applications. [Theory and applications]. MR MR697382 (85a:46001)

[7] A. Castro; J. Cossio; J.M. Neuberger A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., Volume 27 (1997) no. 4, pp. 1041-1053 MR MR1627654 (99f:35056)

[8] B. Gidas; W.M. Ni; L. Nirenberg Symmetry and related properties via the maximum principle, Comm. Math. Phys., Volume 68 (1979) no. 3, pp. 209-243 MR MR544879 (80h:35043)

[9] C. Grumiau, C. Troestler, Symmetries of least energy nodal solutions of Lane–Emden problems on radial domains, accepted for publication in the proceedings of the conference “Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems”, 2007

[10] M. Hoffmann-Ostenhof; T. Hoffmann-Ostenhof; N. Nadirashvili The nodal line of the second eigenfunction of the Laplacian in R2 can be closed, Duke Math. J., Volume 90 (1997) no. 3, pp. 631-640 MR MR1480548 (98m:35146)

[11] K. Jänich Topology, Springer, 1984 MR MR734483 (85a:54001)

[12] A.D. Melas On the nodal line of the second eigenfunction of the Laplacian in R2, J. Differential Geom., Volume 35 (1992) no. 1, pp. 255-263 MR MR1152231 (93g:35100)

[13] J.M. Neuberger A numerical method for finding sign-changing solutions of superlinear Dirichlet problems, Nonlinear World, Volume 4 (1997) no. 1, pp. 73-83 MR MR1452506 (98c:65200)

[14] F. Pacella; T. Weth Symmetry of solutions to semilinear elliptic equations via Morse index, Proc. Amer. Math. Soc., Volume 135 (2007) no. 6, pp. 1753-1762 (electronic). MR MR2286085

[15] M. Willem Principes d'analyse fonctionnelle, Cassini, 2007

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