In this Note, we consider the Lane–Emden problem with Dirichlet boundary conditions, where the domain Ω is an open bounded subset of , is the second eigenvalue of −Δ, and . We prove that, if Ω is 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 avec conditions au bord de Dirichlet, où est ouvert borné, la deuxième valeur propre de −Δ et . Nous prouvons que, sur un convexe de classe , 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.
Accepted:
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@article{CRMATH_2009__347_13-14_767_0, author = {Grumiau, Christopher and Troestler, Christophe}, title = {Nodal line structure of least energy nodal solutions for {Lane{\textendash}Emden} problems}, journal = {Comptes Rendus. Math\'ematique}, pages = {767--771}, publisher = {Elsevier}, volume = {347}, number = {13-14}, year = {2009}, doi = {10.1016/j.crma.2009.04.023}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.04.023/} }
TY - JOUR AU - Grumiau, Christopher AU - Troestler, Christophe TI - Nodal line structure of least energy nodal solutions for Lane–Emden problems JO - Comptes Rendus. Mathématique PY - 2009 SP - 767 EP - 771 VL - 347 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.04.023/ DO - 10.1016/j.crma.2009.04.023 LA - en ID - CRMATH_2009__347_13-14_767_0 ER -
%0 Journal Article %A Grumiau, Christopher %A Troestler, Christophe %T Nodal line structure of least energy nodal solutions for Lane–Emden problems %J Comptes Rendus. Mathématique %D 2009 %P 767-771 %V 347 %N 13-14 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.04.023/ %R 10.1016/j.crma.2009.04.023 %G en %F CRMATH_2009__347_13-14_767_0
Grumiau, Christopher; Troestler, Christophe. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.04.023/
[1] 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] 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] Dual variational methods in critical point theory and applications, J. Funct. Anal., Volume 14 (1973), pp. 349-381 MR MR0370183 (51 #6412)
[4] Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., Volume 96 (2005), pp. 1-18 (MR MR2177179)
[5] 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 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] 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] The nodal line of the second eigenfunction of the Laplacian in can be closed, Duke Math. J., Volume 90 (1997) no. 3, pp. 631-640 MR MR1480548 (98m:35146)
[11] Topology, Springer, 1984 MR MR734483 (85a:54001)
[12] On the nodal line of the second eigenfunction of the Laplacian in , J. Differential Geom., Volume 35 (1992) no. 1, pp. 255-263 MR MR1152231 (93g:35100)
[13] 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] 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] Principes d'analyse fonctionnelle, Cassini, 2007
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