Hostname: page-component-7b9c58cd5d-bslzr Total loading time: 0.001 Render date: 2025-03-15T19:41:28.889Z Has data issue: false hasContentIssue false
  • English
  • Français

On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

Part of: Elliptic equations and systems Partial differential equations Optimality conditions

Published online by Cambridge University Press:  15 January 2019

Vladimir Bobkov  and
Sergey Kolonitskii
Vladimir Bobkov
Affiliation:
Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, Plzeñ 306 14, Czech Republic (bobkov@kma.zcu.cz)
Sergey Kolonitskii
Affiliation:
St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia (s.kolonitsky@spbu.ru)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

Keywords

p-Laplaciansuperlinearsecond eigenvalueleast energy nodal solutionnodal setPayne conjecturepolarization

MSC classification

Primary: 35J92: Quasilinear elliptic equations with $p$-Laplacian
Secondary: 35B06: Symmetries, invariants, etc. 49K30: Optimal solutions belonging to restricted classes
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1163 - 1173
Copyright
Copyright © Royal Society of Edinburgh 2019 

References

1 Aftalion, A. and Pacella, F.. Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains. Comptes Rendus Mathematique 339 (2004), 339–344 (doi: 10.1016/j.crma.2004.07.004).Google Scholar
2Alessandrini, G.. Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains. Commentarii Mathematici Helvetici 69 (1994), 142–154 (doi: 10.1007/BF02564478).Google Scholar
3Anoop, T. V., Drábek, P. and Sasi, S.. On the structure of the second eigenfunctions of the p-Laplacian on a ball. Proc. Am. Math. Soc. 144 (2016), 2503–2512 (doi: 10.1090/proc/12902).Google Scholar
4Bartsch, T., Weth, T. and Willem, M.. Partial symmetry of least energy nodal solutions to some variational problems. J. d'Analyse Mathématique 96 (2005), 1–18 (doi: 10.1007/bf02787822).Google Scholar
5Bobkov, V. E.. On existence of nodal solution to elliptic equations with convex-concave nonlinearities. Ufa Mathematical Journal 5 (2013), 18–30 (doi: 10.13108/2013-5-2-18).Google Scholar
6Bobkov, V. and Kolonitskii, S.. On qualitative properties of solutions for elliptic problems with the p-Laplacian through domain perturbations. (2017). arXiv preprint arXiv:1701.07408.Google Scholar
7Bonheure, D., Bouchez, V., Grumiau, C. and Van Schaftingen, J.. Asymptotics and symmetries of least energy nodal solutions of Lane-Emden problems with slow growth. Commun. Contemp. Math. 10 (2008), 609–631 (doi: 10.1142/S0219199708002910).Google Scholar
8Brock, F. and Solynin, A.. An approach to symmetrization via polarization. Trans. Am. Math. Soc. 352 (2000), 1759–1796 (doi: 10.1090/s0002-9947-99-02558-1).Google Scholar
9Castro, A., Cossio, J. and Neuberger, J. M.. A sign-changing solution for a superlinear Dirichlet problem. Rocky Mountain J. Math. 27 (1997), 1041–1053 (doi: 10.1216/rmjm/1181071858).Google Scholar
10Cuesta, M., De Figueiredo, D. and Gossez, J. P.. The beginning of the Fučik spectrum for the p-Laplacian. J. Differ. Equ. 159 (1999), 212–238 (doi: 10.1006/jdeq.1999.3645).Google Scholar
11Cuesta, M., De Figueiredo, D. G. and Gossez, J. P.. A nodal domain property for the p-Laplacian. Comptes Rendus de l'Académie des Sciences-Series I-Mathematics 330 (2000), 669–673 (doi: 10.1016/S0764-4442(00)00245-7).Google Scholar
12Damascelli, L.. On the nodal set of the second eigenfunction of the laplacian in symmetric domains in ${\open {R}}^N$. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11 (2000), 175–181, http://eudml.org/doc/252373.Google Scholar
13Drábek, P. and Robinson, S. B.. Resonance problems for the p-Laplacian. J. Funct. Anal. 169 (1999), 189–200 (doi: 10.1006/jfan.1999.3501).Google Scholar
14Drábek, P., Kufner, A. and Nicolosi, F.. Quasilinear elliptic equations with degenerations and singularities, vol. 5 (Berlin: Walter de Gruyter, 1997), (doi: 10.1515/9783110804775).Google Scholar
15Fournais, S.. The nodal surface of the second eigenfunction of the Laplacian in ${\open {R}}^D$ can be closed. J. Differ. Equ. 173 (2001), 145–159 (doi: 10.1006/jdeq.2000.3868).Google Scholar
16Ghoussoub, N.. Duality and perturbation methods in critical point theory vol. 107 (Cambridge: Cambridge University Press, 1993).Google Scholar
17Grossi, M., Grumiau, C. and Pacella, F.. Lane-Emden problems: Asymptotic behavior of low energy nodal solutions. Annales de l'Institut Henri Poincare (C) Non Linear Analysis 30 (2013), 121–140 (doi: 10.1016/j.anihpc.2012.06.005).Google Scholar
18Grumiau, C. and Parini, E.. On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian. Archiv der Mathematik 91 (2008), 354–365 (doi: 10.1007/s00013-008-2854-y).Google Scholar
19Grumiau, C. and Troestler, C.. Nodal line structure of least energy nodal solutions for Lane-Emden problems. Comptes Rendus Mathematique 347 (2009), 767–771 (doi: 10.1016/j.crma.2009.04.023).Google Scholar
20Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Nadirashvili, N.. The nodal line of the second eigenfunction of the Laplacian in ${\open {R}}^2$ can be closed. Duke Math. J. 90 (1997), 631–640 (doi: 10.1215/S0012-7094-97-09017-7).Google Scholar
21Kawohl, B.. Rearrangements and convexity of level sets in PDE. Lecture notes in mathematics, vol. 1150 (Berlin, Heidelberg: Springer, 1985 (doi: 10.1007/BFb0075060).Google Scholar
22Lieberman, G. M.. Boundary regularity for solutions of degenerate elliptic equations. Nonlinear. Anal. Theory. Methods. Appl. 12 (1988), 1203–1219 (doi: 10.1016/0362-546x(88)90053-3).Google Scholar
23Nazarov, A. I.. On Solutions to the Dirichlet problem for an equation with p-Laplacian in a spherical layer. Proc. St. Petersburg Math. Soc. 10 (2004), 33–62 (doi: 10.1090/trans2/214/03).Google Scholar
24Payne, L. E.. Isoperimetric inequalities and their applications. SIAM Rev. 9 (1967), 453–488 (doi: 10.1137/1009070).Google Scholar
25Payne, L. E.. On two conjectures in the fixed membrane eigenvalue problem. Zeitschrift für angewandte Mathematik und Physik 24 (1973), 721–729 (doi: 10.1007/BF01597076).Google Scholar
26Serrin, J.. A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43 (1971), 304–318 (doi: 10.1007/BF00250468).Google Scholar
27Tolksdorf, P.. Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51 (1984), 126–150 (doi: 10.1016/0022-0396(84)90105-0).Google Scholar
28Vázquez, J. L.. A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984), 191–202 (doi: 10.1007/bf01449041).Google Scholar