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CONTINUITY OF THE VARIATIONAL EIGENVALUES OF THE p-LAPLACIAN WITH RESPECT TO p

Part of: Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  15 March 2011

ENEA PARINI
ENEA PARINI*
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany (email: eparini@math.uni-koeln.de)
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Abstract

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In this note it is shown that a result of Champion and De Pascale [‘Asymptotic behavior of nonlinear eigenvalue problems involving p-Laplacian type operators’, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 1179–1195] implies that the variational eigenvalues of the p-Laplacian are continuous with respect to p.

Keywords

nonlinear eigenvaluesp-Laplacian

MSC classification

Secondary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 376 - 381
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Anane, A. and Tsouli, N., ‘On the second eigenvalue of the p-Laplacian’, in: Nonlinear Partial Differential Equations (Fés, 1994), Pitman Research Notes in Mathematics Series 343 (1996), pp. 1–9.Google Scholar
[2]Champion, T. and De Pascale, L., ‘Asymptotic behavior of nonlinear eigenvalue problems involving p-Laplacian type operators’, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 1179–1195.CrossRefGoogle Scholar
[3]Garcia Azorero, J. P. and Peral, I., ‘Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues’, Comm. Partial Differential Equations 12 (1987), 1389–1430.Google Scholar
[4]Huang, Y. X., ‘On the eigenvalues of the p-Laplacian with varying p’, Proc. Amer. Math. Soc. 125 (1997), 3347–3354.CrossRefGoogle Scholar
[5]Juutinen, P. and Lindqvist, P., ‘On the higher eigenvalues for the ∞-eigenvalue problem’, Calc. Var. Partial Differential Equations 23 (2005), 169–192.CrossRefGoogle Scholar
[6]Juutinen, P., Lindqvist, P. and Manfredi, J. J., ‘The ∞-eigenvalue problem’, Arch. Ration. Mech. Anal. 148 (1999), 89–105.Google Scholar
[7]Kawohl, B. and Fridman, V., ‘Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant’, Comment. Math. Univ. Carolin. 44 (2003), 659–667.Google Scholar
[8]Lê, A. and Schmitt, K., ‘Variational eigenvalues of degenerate eigenvalue problems for the weighted p-Laplacian’, Adv. Nonlinear Stud. 5 (2005), 573–585.CrossRefGoogle Scholar
[9]Parini, E., ‘The second eigenvalue of the p-Laplacian as p goes to 1’, Int. J. Differ. Equ. (2010), Article ID 984671.Google Scholar