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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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