Global bifurcation in generic systems of nonlinear Sturm-Liouville problems

Abstract:

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