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

Summary: The author considers a system of coupled nonlinear Sturm-Liouville boundary value problems

L 1 u:=-(p 1 u ' ) ' +q 1 u=μu+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=νv+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 μ, ν are real spectral parameters. It is shown that if the functions f and g are ‘generic’ then for all integers m,n0, there are smooth 2-dimensional manifolds 𝒮 m 1 , 𝒮 n 2 , of ‘semi-trivial’ solutions to the system which bifurcate from the eigenvalues μ m , ν n , of L 1 , L 2 , respectively. Furthermore, there are smooth curves mn 1 𝒮 m 1 , mn 2 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, 𝒩 mn , which ‘links’ the curves mn 1 , mn 2 . Nodal properties of solutions on 𝒩 mn and global properties of 𝒩 mn are discussed.

34B24Sturm-Liouville theory
34B15Nonlinear boundary value problems for ODE
58E07Abstract bifurcation theory
34C23Bifurcation (ODE)