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On the Principal Eigencurve of the p-Laplacian: Stability Phenomena

Published online by Cambridge University Press:  20 November 2018

Abdelouahed El Khalil ,
Said El Manouni  and
Mohammed Ouanan
Abdelouahed El Khalil
Affiliation:
Département de Mathématiques et de Génie Industriel, École Polytechnique de Montréal, Montréal, [email protected]
Said El Manouni
Affiliation:
Department of Mathematics, Faculty of Sciences Dhar-Mahraz, P.O. Box 1796 Atlas, Fez 30000, [email protected][email protected]
Mohammed Ouanan
Affiliation:
Department of Mathematics, Faculty of Sciences Dhar-Mahraz, P.O. Box 1796 Atlas, Fez 30000, [email protected][email protected]
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Abstract

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We show that each point of the principal eigencurve of the nonlinear problem

$$-{{\Delta }_{p}}u-\text{ }\lambda m(x){{\left| u \right|}^{p-2}}u=\mu {{\left| u \right|}^{p-2}}u\,\,\text{in}\Omega ,$$

is stable (continuous) with respect to the exponent $p$ varying in $\left( 1,\infty \right)$; we also prove some convergence results of the principal eigenfunctions corresponding.

Keywords

35P3035P6035J70p-Laplacian with indefinite weightprincipal eigencurveprincipal eigenvalueprincipal eigenfunctionstability
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 3 , 01 September 2006 , pp. 358 - 370
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Adams, R., Sobolev Spaces, Pure and Applied Mathematics 65, Academic Press, New York, 1975.Google Scholar
[2] Anane, A., Simplicité et isolation de la première valeur propre du p-Laplacien avec poids. C. R. Acad. Sci. Paris, Sér. I Math. 305(1987), no. 16, 725–728.Google Scholar
[3] Atkinson, C. and Champion, C. R., Some-boundary value problems for the equation ∇ · (| ∇ϕ|N = 0 . Quart. J. Mech. Appl. Math. 37(1984), no. 3, 401–419.Google Scholar
[4] Atkinson, C. and Jones, C. W., Similarity solutions in some nonlinear diffusion problems and in boundary-layer flow of a pseudo plastic fluid. Quart. J. Mech. Appl. Math. 27(1974), 193–211.Google Scholar
[5] Azorero, J. P. G. and Alonso, I. P., Existence and uniqueness for the p-Laplacian: nonlinear eigenvalues. Comm. Partial Differential Equations 12(1987), no. 12, 1389–1430.Google Scholar
[6] Barenblatt, G. I., On self similar notions of compressible fluids in a porous medium. Prikl. Mat. Mekh. 16(1952), 679–698 (in Russian).Google Scholar
[7] Binding, P. A. and Huang, Y. X., The principal eigencurve for the p-Laplacian. Differential Integral Equations 8(1995), no. 2, 405–414.Google Scholar
[8] Brezis, H., Analyse fonctionnelle. Théorie et applications. CollectionMathématiques Appliquées pour la Maitrise. Masson, Paris, 1983.Google Scholar
[9] Courant, R. and Hilbert, D., Methods of Mathematical Physics. Vol. 1, Interscience Publishers, New York, 1953.Google Scholar
[10] Diaz, J. I., Nonlinear partial differential equations and free boundaries. Vol. 1, Elliptic Equations, Research Notes in Mathematics 106, Pitman, Boston, 1985.Google Scholar
[11] Dibenedetto, E., C 1+α -local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(1983), no. 8, 827–850.Google Scholar
[12] Edmunds, D. E. and Evans, W. D., Spectral Theory and Differential Operators. Clarendon Press, Oxford, 1990.Google Scholar
[13] El Khalil, A., Autour de la première courbe propre du p-Laplacien. Thèse de doctorat, Faculté des Sciences Dhar-Mahraz, Fès, 1999.Google Scholar
[14] El Khalil, A., Lindqvist, P., and Touzani, A., On the stability of the first eigenvalue of Apu + λ g (x) | u |p–2 u = 0 with varying p. Rend. Mat. Appl. 24(2004), no. 2, 321–336.Google Scholar
[15] El Khalil, A., Lindqvist, P., and Touzani, A., On the first eigenvalue of the p-Laplacian. In: Partial Differential Equations, Lecture Notes in Pure and Applied Mathematics 229, Dekker, New York, 2002, pp. 195–205.Google Scholar
[16] Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften 224, Springer-Verlag, Berlin, 1983.Google Scholar
[17] Hess, P. and Kato, T., On some linear and nonlinear eigenvalue problems with an indefinite weight function. Comm. Partial Differential Equations 5(1980), no. 10, 999–1030.Google Scholar
[18] Lindqvist, P., On the equation div(|∇ u|p–2 ∇u) + λ|u|p–2 u = 0 . Proc. Amer. Math. Soc. 109(1990), no. 1, 157–164.Google Scholar
[19] Lindqvist, P., On nonlinear Rayleigh quotients. Potential Anal. 2(1993), no. 3, 199–218.Google Scholar
[20] Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969.Google Scholar
[21] Otani, M. and Teshima, T., On the first eigenvalue of some quasilinear elliptic equations. Proc. Japan Acad. Sci. Ser. A. Math. Sci. 64(1988), no. 1, 8–10.Google Scholar
[22] Pelissier, M. C. and Reynaud, L., Étude d’un modèle mathématique d’écoulement de glacier. C. R. Acad. Sci. Paris Sér. A 279(1974), 531–534.Google Scholar
[23] Philip, J. R., n-diffusion. Austral. J. Phys. 14(1961), 1–13.Google Scholar
[24] Showalter, R. E., Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs 49, American Mathematical Society, Providence, RI, 1997.Google Scholar
[25] Smagorinsky, J., General circulation experiments with the primitive equations. I: The basic experiment. Monthly Wea. Rev. 91(1963), 99–152.Google Scholar
[26] Vázquez, J. L., A strong maximum principle for some quasilinear elliptic equations. App. Math. Optim. 12(1984), no. 3, 191–202.Google Scholar