Global Bifurcation in Generic Systems of Nonlinear Sturm-liouville Problems

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 = νv + vg(·, u, v), in (0, 1), a20v(0) + b20v′(0) = 0, a21v(1) + b21v′(1) = 0, where μ, ν 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 S1 m, S2 n, of ‘semi-trivial’ solutions of the system which bifurcate from the eigenvalues μm, νn, of L1, L2, respectively. Furthermore, there are smooth curves B1 mn ⊂ S1 m, B2 mn ⊂ S2 n, 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 B1 mn, B2 mn. Nodal properties of solutions on Nmn and global properties of Nmn are also discussed.