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The structure of Rabinowitz' global bifurcating continua for problems with weak nonlinearities

Part of: Partial differential equations

Published online by Cambridge University Press:  26 February 2010

Rehana Bari  and
Bryan P. Rynne
Rehana Bari
Affiliation:
Department of Mathematics, University of Dhaka, Bangladesh.
Bryan P. Rynne
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland
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Abstract

Rabinowitz' global bifurcation theorem shows that for a large class of nonlinear eigenvalue problems a continuum (i.e., a closed, connected set) of solutions bifurcates from the trivial solution at each eigenvalue (or characteristic value) of odd multiplicity of the linearized problem (linearized at the trivial solution). Each continuum must either be unbounded, or must meet some other eigenvalue. This paper considers a class of such nonlinear eigenvalue problems having simple eigenvalues and a “weak” nonlinear term. A result regarding the location of the continua is obtained which shows, in particular, that in this case the bifurcating continua must be unbounded. Also, under further differentiability conditions it is shown that the continua are smooth, 1-dimensional curves and that there are no non-trivial solutions of the equation other than those lying on the bifurcating continua.

MSC classification

Secondary: 35B32: Bifurcation
Type
Research Article
Information
Mathematika , Volume 44 , Issue 2 , December 1997 , pp. 419 - 433
Copyright
Copyright © University College London 1997

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References

1.Agmon, S.. Lectures on Elliptic Boundary Value Problems (Van Nostrand, 1965).Google Scholar
2.Chiappinelli, R.. On eigenvalues and bifurcation for nonlinear Sturm Liouville operators. Bollettino U. M. I., 4A (1985), 7783.Google Scholar
3.Cosner, C.. Bifurcation from higher eigenvalues in nonlinear elliptic equations: continua that meet infinity. Nonlinear Anal, Theory, Methods and Applications, 12 (1988), 271277.CrossRefGoogle Scholar
4.Crandall, M. G. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Fund. Anal., 8 (1971). 321340.CrossRefGoogle Scholar
5.Dancer, E. N.. A note on bifurcation from infinity. Quart. J. Math. Oxford, 25 (1974), 8184.CrossRefGoogle Scholar
6.Deimling, K.. Nonlinear Functional Analysis (Springer, 1985).CrossRefGoogle Scholar
7.Kato, T.. Perturbation Theory for Linear Operators (Springer, 1984).Google Scholar
8.Healey, T. and Kielhōfer, H.. Preservation of nodal structure on global bifurcating solution branches of elliptic equations with symmetry. J. Diff. Equations, 106 (1993), 70–89.CrossRefGoogle Scholar
9.Holzmann, M. and Kielhōfer, H.. Uniqueness of global positive solution branches of nonlinear elliptic problems. Math. Ann., 300 (1994), 221241.CrossRefGoogle Scholar
10.Kielhōfer, H.. Smoothness of global positive branches of nonlinear elliptic problems over symmetrical domains. Math. Zeitschrift, 211 (1992), 4148.CrossRefGoogle Scholar
11.Naimark, M. A.. Linear Differential Operators, Part I (Harrap, 1968).Google Scholar
12.Rabinowitz, P. H.. Some global results for nonlinear eigenvalue problems. J. Fund. Analysis, 7 (1971), 487513.CrossRefGoogle Scholar
13.Schmitt, K. and Smith, H. L.. On eigenvalue problems for nondifferentiable mappings. J. Diff. Equations, 33 (1979), 294–319.CrossRefGoogle Scholar
14.Uhlenbeck, K.. Generic properties of eigenfunctions. Am. J. Math., 98 (1976), 10591078.CrossRefGoogle Scholar
15.Zeidler, E.. Nonlinear Functional Analysis and its Applications, Vol I (Springer, 1986).CrossRefGoogle Scholar