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International Journal of Differential Equations
Volume 2010 (2010), Article ID 984671, 23 pages
http://dx.doi.org/10.1155/2010/984671
Research Article

The Second Eigenvalue of the -Laplacian as Goes to

Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D 50931 Köln , Germany

Received 15 July 2009; Accepted 29 September 2009

Academic Editor: Norimichi Hirano

Copyright © 2010 Enea Parini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The asymptotic behaviour of the second eigenvalue of the -Laplacian operator as goes to 1 is investigated. The limit setting depends only on the geometry of the domain. In the particular case of a planar disc, it is possible to show that the second eigenfunctions are nonradial if is close enough to 1.