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On the structure of the second eigenfunctions of the $ p$-Laplacian on a ball


Authors: T. V. Anoop, P. Drábek and Sarath Sasi
Journal: Proc. Amer. Math. Soc. 144 (2016), 2503-2512
MSC (2010): Primary 35J92, 35P30, 35B06, 49R05
DOI: https://doi.org/10.1090/proc/12902
Published electronically: October 19, 2015
MathSciNet review: 3477066
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Abstract: In this paper, we prove that the second eigenfunctions of the $ p$-Laplacian, $ p>1$, are not radial on the unit ball in $ \mathbb{R}^N,$ for any $ N\ge 2.$ Our proof relies on the variational characterization of the second eigenvalue and a variant of the deformation lemma. We also construct an infinite sequence of eigenpairs $ \{\tau _n,\Psi _n\}$ such that $ \Psi _n$ is nonradial and has exactly $ 2n$ nodal domains. A few related open problems are also stated.


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Additional Information

T. V. Anoop
Affiliation: Department of Mathematics, Indian Institute of Technology Madras, Chennai 36, India
Email: anoop@iitm.ac.in

P. Drábek
Affiliation: Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 306 14 Plzeň, Czech Republic
Email: pdrabek@kma.zcu.cz

Sarath Sasi
Affiliation: School of Mathematical Sciences, National Institute for Science Education and Research, Jatni, Odisha, PIN 752050, India
Email: sarath@niser.ac.in

DOI: https://doi.org/10.1090/proc/12902
Keywords: $p$-Laplacian, nonlinear eigenvalue problem, symmetry properties, shape derivative, variational characterization
Received by editor(s): January 22, 2015
Received by editor(s) in revised form: June 19, 2015, and July 9, 2015
Published electronically: October 19, 2015
Additional Notes: The work of the first and third authors was funded by the project EXLIZ - CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic.
The second author was supported by the Grant Agency of Czech Republic, Project No. 13-00863S
The second author is the corresponding author
The first and third authors acknowledge the support received from the Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia during the period in which this work was done.
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society