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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Global bifurcation in generic systems
of nonlinear Sturm-Liouville problems


Author: Bryan P. Rynne
Journal: Proc. Amer. Math. Soc. 127 (1999), 155-165
MSC (1991): Primary 34B15; Secondary 34B24, 58E07
MathSciNet review: 1487336
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Abstract: We consider the system of coupled nonlinear Sturm-Liouville boundary value problems $ \begin{array}{c} L_1 u := -(p_1 u')' + q_1 u = \mu u + u f(\cdot,u,v), \quad \text{in}\ (0,1),\\ [1 ex] a_{10} u(0) + b_{10} u'(0) = 0, \quad a_{11} u(1) + b_{11} u'(1) = 0, \end{array} $ $ \begin{array}{c} L_2 v := -(p_2 v')' + q_2 v = \nu v + v g(\cdot,u,v), \quad \text{in}\ (0,1),\\ [1 ex] a_{20} v(0) + b_{20} v'(0) = 0, \quad a_{21} v(1) + b_{21} v'(1) = 0, \end{array} $

where $\mu$, $\nu$ are real spectral parameters. It will be shown that if the functions $f$ and $g$ are `generic' then for all integers $m,\,n \ge 0$, there are smooth 2-dimensional manifolds $ {\mathcal S}_m^1$, $ {\mathcal S}_n^2$, of `semi-trivial' solutions of the system which bifurcate from the eigenvalues $\mu _m$, $\nu _n$, of $L_1$, $L_2$, respectively. Furthermore, there are smooth curves $ {\mathcal B}_{mn}^1 \subset {\mathcal S}_m^1$, $ {\mathcal B}_{mn}^2 \subset {\mathcal S}_n^2$, along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of `non-trivial' solutions. It is shown that there is a single such manifold, $ {\mathcal N}_{mn}$, which `links' the curves $ {\mathcal B}_{mn}^1$, $ {\mathcal B}_{mn}^2$. Nodal properties of solutions on $ {\mathcal N}_{mn}$ and global properties of $ {\mathcal N}_{mn}$ are also discussed.


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Additional Information

Bryan P. Rynne
Affiliation: Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland
Email: bryan@ma.hw.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04763-2
PII: S 0002-9939(99)04763-2
Keywords: Global bifurcation, genericity, Sturm-Liouville systems
Received by editor(s): May 2, 1997
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society