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Global bifurcation for 2mth-order boundary value problems and infinitely many solutions of superlinear problems. (English) Zbl 1029.34015

The author studies a boundary value problem associated to a 2mth-order ordinary differential equation of the form

Lu(x)=p(x)u(x)+g(x,u (0) (x),,u (2m-1) (x))u(x),

where L is a selfadjoint, disconjugate operator on [0,π] and the boundary conditions are separated. It is assumed that g is “superlinear at infinity” and that lim |ξ|0 g(x,ξ)=0. It is proved the existence of infinitely many solutions having specified nodal properties.

The main result represents a generalization to higher-order problems of a result by P. Hartman [J. Differ. Equations 26, 37-53(1997; Zbl 0365.34032)]. In the proof, it is used a generalization of the Rabinowitz global bifurcation theorem together with general results on the nodal properties of the solutions to the linear eigenvalue problem Lu=μpu.

Related results for fourth-order problems have been given, among others, by M. Conti, S. Terracini and G. Verzini [Infinitely many solutions to fourth order superlinear periodic problems (preprint)] and by M. Henrard and F. Sadyrbaev [Nonlinear Anal., Theory Methods Appl. 33, 281-302 (1998; Zbl 0937.34020)].

34B15Nonlinear boundary value problems for ODE
47J15Abstract bifurcation theory
34C23Bifurcation (ODE)