Proceedings of the Royal Society of Edinburgh: Section A Mathematics



Asymptotic behaviour of nonlinear eigenvalue problems involving $p$-Laplacian-type operators


Thierry Champion a1 and Luigi De Pascale a2
a1 Laboratoire d'Analyse Non Linéaire Appliquée, UFR des Sciences et Techniques, Université du Sud Toulon-Var, Avenue de l'Université, BP 20132, 83957 La Garde Cedex, France
a2 Dipartimento di Matematica Applicata, Università di Pisa, Via Buonarroti 1/c, 56127 Pisa, Italy (depascal@dm.unipi.it)

Article author query
champion t   [Google Scholar] 
de pascale l   [Google Scholar] 
 

Abstract

We study the asymptotic behaviour of two nonlinear eigenvalue problems which involve $p$-Laplacian-type operators. In the first problem we consider the limit as $p\to\infty$ of the sequences of the $k$th eigenvalues of the $p$-Laplacian operators. The second problem we study is the homogenization of nonlinear eigenvalue problems for some $p$-Laplacian-type operators with $p$ fixed. Our asymptotic analysis relies on a convergence result for particular critical values of a class of Rayleigh quotients, stated in a unified framework, and on the notion of $\varGamma$-convergence.

(Published Online December 3 2007)
(Received May 25 2006)
(Accepted October 25 2006)