### Abstract

We consider the system of coupled nonlinear Sturm-Liouville boundary value problems L_{1}u := -(p_{1}u')' + q_{1}u = µu + uf(-, u, v), in (0,1), a10u(0) + b_{10}u'(0) = 0, a1_{1}u(1) + b_{11}u'(1) = 0, L_{2}v := -(P_{2}v')' + q_{2}v = vv + vg(-, u, v), in (0, 1), a_{20}v(0) + b_{20}v'(0) = 0, a_{21}v(1) + b_{21}v'(1) = 0, where µ, v are real spectral parameters. It will be shown that if the functions f and g are 'generic' then for all integers m, n = 0, there are smooth 2-dimensional manifolds S_{m}^{1}, S_{n}^{2}, of 'semi-trivial' solutions of the system which bifurcate from the eigenvalues µ_{m}, v_{n}, of L_{1}, L_{2}, respectively. Furthermore, there are smooth curves B_{mn}^{1} ? S_{m}^{1}, B_{mn}^{2} ? S_{n}^{2}, along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of 'nontrivial' solutions. It is shown that there is a single such manifold, N_{mn}, which 'links' the curves B_{mn}^{1}, B_{mn}^{2}. Nodal properties of solutions on N_{mn} and global properties of N_{mn} are also discussed. © 1999 American Mathematical Society.

Original language | English |
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Pages (from-to) | 155-165 |

Number of pages | 11 |

Journal | Proceedings of the American Mathematical Society |

Volume | 127 |

Issue number | 1 |

Publication status | Published - 1999 |

### Keywords

- Genericity
- Global bifurcation
- Sturm-liouville systems