Global bifurcation for 2\(m\)th-order boundary value problems and infinitely many solutions of superlinear problems
The author studies a boundary value problem associated to a \(2m\)th-order ordinary differential equation of the form \[ Lu(x)=p(x)u(x) + g(x,u^{(0)}(x), \dots, u^{(2m-1)}(x))u(x), \] where \(L\) is a selfadjoint, disconjugate operator on \([0,\pi]\) and the boundary conditions are separated. It is assumed that \(g\) is ``superlinear at infinity'' and that \(\lim_{|\xi|\to 0} g(x,\xi)=0\). 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 \textit{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 \(Lu=\mu p u\). Related results for fourth-order problems have been given, among others, by \textit{M. Conti, S. Terracini} and \textit{G. Verzini} [Infinitely many solutions to fourth order superlinear periodic problems (preprint)] and by \textit{M. Henrard} and \textit{F. Sadyrbaev} [Nonlinear Anal., Theory Methods Appl. 33, 281-302 (1998; Zbl 0937.34020)].
scientific article; zbMATH DE number 1902857