Global bifurcation in generic systems of nonlinear Sturm-Liouville problems

We consider the system of coupled nonlinear Sturm-Liouville boundary value problems $\begin{array}{c} L_1 u := -(p_1 u')' + q_1 u = \mu u + u f(\cdot,u,v), \quad \text{in} (0,1), [1 ex] a_{10} u(0) + b_{10} u'(0) = 0, \quad a_{11} u(1) + b_{11} u'(1) = 0, \end{array}$ $\begin{array}{c} L_2 v := -(p_2 v')' + q_2 v = \nu v + v g(\cdot,u,v), \quad \text{in} (0,1), [1 ex] a_{20} v(0) + b_{20} v'(0) = 0, \quad a_{21} v(1) + b_{21} v'(1) = 0, \end{array}$

where $\mu$, $\nu$ are real spectral parameters. It will be shown that if the functions $f$ and $g$ are `generic' then for all integers $m,\,n \ge 0$, there are smooth 2-dimensional manifolds ${\mathcal S}_m^1$, ${\mathcal S}_n^2$, of `semi-trivial' solutions of the system which bifurcate from the eigenvalues $\mu _m$, $\nu _n$, of $L_1$, $L_2$, respectively. Furthermore, there are smooth curves ${\mathcal B}_{mn}^1 \subset {\mathcal S}_m^1$, ${\mathcal B}_{mn}^2 \subset {\mathcal S}_n^2$, 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, ${\mathcal N}_{mn}$, which `links' the curves ${\mathcal B}_{mn}^1$, ${\mathcal B}_{mn}^2$. Nodal properties of solutions on ${\mathcal N}_{mn}$ and global properties of ${\mathcal N}_{mn}$ are also discussed.