Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


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
Published electronically: October 19, 2015
MathSciNet review: 3477066
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Jiří Benedikt, Pavel Drábek, and Petr Girg, The second eigenfunction of the $ p$-Laplacian on the disk is not radial, Nonlinear Anal. 75 (2012), no. 12, 4422-4435. MR 2927111,
  • [2] A. Chorwadwala and R. Mahadevan, A shape optimization problem for the $ p$-laplacian, ArXiv e-prints (accepted in Proceedings of Royal Society of Edinburgh).
  • [3] Manuel A. del Pino and Raúl F. Manásevich, Global bifurcation from the eigenvalues of the $ p$-Laplacian, J. Differential Equations 92 (1991), no. 2, 226-251. MR 1120904 (92g:35028),
  • [4] Pavel Drábek and Jaroslav Milota, Methods of nonlinear analysis. Applications to differential equations, 2nd ed., Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer Basel AG, Basel, 2013. MR 3025694
  • [5] Pavel Drábek and Stephen B. Robinson, Resonance problems for the $ p$-Laplacian, J. Funct. Anal. 169 (1999), no. 1, 189-200. MR 1726752 (2000j:35096),
  • [6] Pavel Drábek and Stephen B. Robinson, On the generalization of the Courant nodal domain theorem, J. Differential Equations 181 (2002), no. 1, 58-71. MR 1900460 (2003g:35170),
  • [7] B. Emamizadeh and M. Zivari-Rezapour, Monotonicity of the principal eigenvalue of the $ p$-Laplacian in an annulus, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1725-1731. MR 2373602 (2009b:35105),
  • [8] J. P. García Azorero and I. Peral Alonso, Existence and nonuniqueness for the $ p$-Laplacian: nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), no. 12, 1389-1430. MR 912211 (89e:35058),
  • [9] Nassif Ghoussoub, Duality and perturbation methods in critical point theory. With appendices by David Robinson, Cambridge Tracts in Mathematics, vol. 107, Cambridge University Press, Cambridge, 1993. MR 1251958 (95a:58021)
  • [10] Bernd Kawohl and Peter Lindqvist, Positive eigenfunctions for the $ p$-Laplace operator revisited, Analysis (Munich) 26 (2006), no. 4, 545-550. MR 2329592 (2008g:35151),
  • [11] Gary M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203-1219. MR 969499 (90a:35098),
  • [12] Peter Lindqvist, On the equation $ {\rm div}\,(\vert \nabla u\vert ^{p-2}\nabla u)+\lambda \vert u\vert ^{p-2}u=0$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157-164. MR 1007505 (90h:35088),
  • [13] Enea Parini, The second eigenvalue of the $ p$-Laplacian as $ p$ goes to $ 1$, Int. J. Differ. Equ. (2010), Art. ID 984671, 23. MR 2575290 (2010m:35357)
  • [14] L. E. Payne, Isoperimetric inequalities and their applications, SIAM Rev. 9 (1967), 453-488. MR 0218975 (36 #2058)
  • [15] Lawrence E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys. 24 (1973), 721-729 (English, with German summary). MR 0333487 (48 #11812)
  • [16] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191-202. MR 768629 (86m:35018),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35J92, 35P30, 35B06, 49R05

Retrieve articles in all journals with MSC (2010): 35J92, 35P30, 35B06, 49R05

Additional Information

T. V. Anoop
Affiliation: Department of Mathematics, Indian Institute of Technology Madras, Chennai 36, India

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

Sarath Sasi
Affiliation: School of Mathematical Sciences, National Institute for Science Education and Research, Jatni, Odisha, PIN 752050, India

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

American Mathematical Society