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Zbl 0905.34021
Rynne, Bryan P.
Global bifurcation in generic systems of nonlinear Sturm-Liouville problems. (English)
[J] Proc. Am. Math. Soc. 127, No.1, 155-165 (1999). ISSN 0002-9939; ISSN 1088-6826/e

Summary: The author considers a system of coupled nonlinear Sturm-Liouville boundary value problems $$L_1 u := -(p_1 u')' + q_1 u = \mu u + u f(\cdot,u,v),\text{ in }(0,1),$$ $$a_{10} u(0) + b_{10} u'(0) = 0,\quad a_{11} u(1) + b_{11} u'(1) = 0,$$ $$L_2 v := -(p_2 v')' + q_2 v = \nu v + v g(\cdot,u,v),\text{ in }(0,1),$$ $$a_{20} v(0) + b_{20} v'(0) = 0,\quad a_{21} v(1) + b_{21} v'(1) = 0,$$ where $\mu$, $\nu$ are real spectral parameters. It is shown that if the functions $f$ and $g$ are `generic' then for all integers $m, n \ge 0$, there are smooth 2-dimensional manifolds ${\cal S}_m^1$, ${\cal S}_n^2$, of `semi-trivial' solutions to the system which bifurcate from the eigenvalues $\mu_m$, $\nu_n$, of $L_1$, $L_2$, respectively. Furthermore, there are smooth curves ${\cal B}_{mn}^1 \subset {\cal S}_m^1$, ${\cal B}_{mn}^2 \subset {\cal S}_n^2$, 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, ${\cal N}_{mn}$, which `links' the curves ${\cal B}_{mn}^1$, ${\cal B}_{mn}^2$. Nodal properties of solutions on ${\cal N}_{mn}$ and global properties of ${\cal N}_{mn}$ are discussed.

MSC 2000:: *34B24 Sturm-Liouville theory
34B15 Nonlinear boundary value problems of ODE
58E07 Abstract bifurcation theory
34C23 Bifurcation (periodic solutions)

Keywords: global bifurcation; genericity; Sturm-Liouville systems; coupled nonlinear

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