We consider the system of coupled nonlinear Sturm-Liouville boundary value problems L1u := -(p1u')' + q1u = µu + uf(-, u, v), in (0,1), a10u(0) + b10u'(0) = 0, a11u(1) + b11u'(1) = 0, L2v := -(P2v')' + q2v = vv + vg(-, u, v), in (0, 1), a20v(0) + b20v'(0) = 0, a21v(1) + b21v'(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 Sm1, Sn2, of 'semi-trivial' solutions of the system which bifurcate from the eigenvalues µm, vn, of L1, L2, respectively. Furthermore, there are smooth curves Bmn1 ? Sm1, Bmn2 ? Sn2, 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, Nmn, which 'links' the curves Bmn1, Bmn2. Nodal properties of solutions on Nmn and global properties of Nmn are also discussed. © 1999 American Mathematical Society.
|Number of pages||11|
|Journal||Proceedings of the American Mathematical Society|
|Publication status||Published - 1999|
- Global bifurcation
- Sturm-liouville systems