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Singular perturbation of a nonlinear problem with multiple solutions

Published online by Cambridge University Press:  14 November 2011

Anne-Marie Lefevere
Anne-Marie Lefevere
Affiliation:
Departement de Mathématiques et d'Informatique, Faculté des Sciences, Université de Pau, 64000 Pau, France
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Synopsis

A nonlinear boundary value problem (P) having positive parameters L and a is considered. We associate with it a family of perturbed problems () affected by the presence of a barrier parameter γ related to L and a. There is a critical value L*(a) of the parameter L such that for L >L*(a), (P) has no regular solution. Then some natural extensions of (P), solutions of a free boundary value problem, arise as singular limits of ().

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 3-4 , 1985 , pp. 327 - 341
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Brauner, C. M. and Nicolaenko, B.. Singular perturbation, multiple solutions and hysteresis in an nonlinear problem. Lecture Notes in Mathematics 594, pp. 50–76 (Berlin: Springer, 1977).Google Scholar
2Brauner, C. M. and Nicolaenko, B.. On nonlinear eigenvalue problems which extend into free boundaries problems. Proc. Conf. on Nonlinear Eigenvalue Problems. Lecture Notes in Mathematics 782, pp. 61–100 (Berlin: Springer, 1979).Google Scholar
3Brauner, C. M. and Nicolaenko, B.. Free boundary value problems as singular limits of nonlinear eigenvalue problems. Proc. Symp. on Free Boundary Problems, Vol. II, pp. 61–84 (Rome: Istituto Nazionale di Alta Matematica, 1980).Google Scholar
4Crandall, M. G. and Rabinowitz, P. H.. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rational Mech. Anal. 52 (1973), 161–180.CrossRefGoogle Scholar
5Crandall, M. G. and Rabinowitz, P. H.. Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Rational Mech. Anal. 58 (1975), 207–218.CrossRefGoogle Scholar
6Frank, L. S. and Groesen, E. W. Van. Singular perturbations of an elliptic operator with discontinuous nonlinearity. Proc. Conf. on Analytical and Numerical Approaches to Asymptotic Problems, pp. 289–303 (Amsterdam: North Holland, 1981).Google Scholar
7Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order (Berlin: Springer, 1977).CrossRefGoogle Scholar
8Guyot, J.. Etude mathématique et numérique d'un problème elliptique non linéaire avec points de retournement. Thèse, Ecole Centrale de Lyon (1981).Google Scholar
9Joseph, D. D.. Stability of Fluid Motions II (Berlin: Springer, 1976).CrossRefGoogle Scholar
10Keener, J. P. and Keller, H. B.. Positive solutions of convex nonlinear eigenvalue problems. J. Differential Equations 16 (1974), 103–125.CrossRefGoogle Scholar
11Lefevere, A. M.. A nonlinear boundary value problem suggested by the Laplace equation for an elastic and axisymmetric membrane. SIAM J. Math. Anal. 13 (1982), 48–60.CrossRefGoogle Scholar
12Lefevere, A. M.. Perturbation singulière pour un problème aux limites non linéaire à solutions multiples. C. R. Acad. Sci. Paris, Ser. I 295 (1982), 231–234.Google Scholar
13Lefevere, A. M.. Perturbation singulière pour un probleme aux limites non lineáire à solutions multiples. Rapport interne du Labo. Anal. Num. de Pau. (France).Google Scholar
14Lions, J. L.. Quelques méthodes de résolution des problèmes aux limites non lineáires (Paris: Dunod, 1969).Google Scholar
15Lions, J. L.. Perturbations singulières dans les problemes aux limites et en contrôle optimal. Lecture Notes in Mathematics 323 (Berlin: Springer, 1973).Google Scholar
16Mignot, F. and Puel, J. P.. Sur une classe de problèmes non linéaires avec non linéarité positive croissante convexe. Comm. Partial Differential Equations 5 (8) (1980), 791–836.CrossRefGoogle Scholar
17Moore, W. J.. Chimie Physique (Paris: Dunod, 1965).Google Scholar
18Puel, J. P.. Existence, comportement à l'infini et stabilité dans certains problèmes quasilinéaires elliptiques et paraboliques d'ordre 2. Ann. Scuola Norm. Sup. Pisa 3 (1976), 85–119.Google Scholar
19Quenisset, J. M.. Chimie du solide. Thèse, Bordeaux, Labo. C.N.R.S. 1980.Google Scholar
20Sattinger, D. H.. Topics in stability and bifurcation theory. Lecture Notes in Mathematics 309 (Berlin: Springer, 1973).Google Scholar
21Stampacchia, G.. Equations elliptiques du second ordre à coefficients discontinus (Presses de l'Université de Montréal, 1966).CrossRefGoogle Scholar