Global bifurcation in generic systems of nonlinear Sturm-Liouville problems

We consider the system of coupled nonlinear Sturm-Liouville boundary value problems L(1)u: = -(p(1)u')' + q(1)u - un + uf(., u,v), in (0,1), a(10)u(0) + b(10)u '(0) = (0), a(11)u(1) + b(11)u'(1) = 0, L(2)v = -(p(2)v')' + q(2)v = upsilo n v + 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 mu, upsilon are real spectral parameters. It will be shown that if th e functions f and g are 'generic' then for all integers m, n greater than o r equal to 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 mu(m), upsilon(n), of L-1, L-2, respectively. Furthermore, there are smooth curves B-mn(1) subset of S-m(1), B-mn(2) subset of S-n(2), along which sec ondary bifurcations take place, giving rise to smooth, a-dimensional manifo lds of 'nontrivial' solutions. It is shown that there isa single such manif old, N-mn, which 'links' the curves B-mn(1), B-mn(2). Nodal properties of s olutions on N-mn and global properties of N-mn are also discussed.