Global bifurcation in generic systems of nonlinear Sturm-Liouville problems. (English) Zbl 0905.34021

Rynne, Bryan P.

Proc. Am. Math. Soc. 127, No. 1, 155-165 (1999).

Summary: The author considers a system of coupled nonlinear Sturm-Liouville boundary value problems \[ L_1 u := -(p_1 u')' + q_1 u = \mu u + u f(\cdot,u,v),\text{ in }(0,1), \]
\[ a_{10} u(0) + b_{10} u'(0) = 0,\quad a_{11} u(1) + b_{11} u'(1) = 0, \]
\[ L_2 v := -(p_2 v')' + q_2 v = \nu v + v g(\cdot,u,v),\text{ in }(0,1), \]
\[ a_{20} v(0) + b_{20} v'(0) = 0,\quad a_{21} v(1) + b_{21} v'(1) = 0, \] where \(\mu\), \(\nu\) are real spectral parameters. It is shown that if the functions \(f\) and \(g\) are ‘generic’ then for all integers \(m, n \geq 0\), there are smooth 2-dimensional manifolds \({\mathcal S}_m^1\), \({\mathcal S}_n^2\), of ‘semi-trivial’ solutions to 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 discussed.

MSC:

34B24	Sturm-Liouville theory
34B15	Nonlinear boundary value problems for ordinary differential equations
58E07	Variational problems in abstract bifurcation theory in infinite-dimensional spaces
34C23	Bifurcation theory for ordinary differential equations

Keywords:

global bifurcation; genericity; Sturm-Liouville systems; coupled nonlinear