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Global bifurcation for 2$m$th-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 $2m$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 {\it 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 {\it M. Conti, S. Terracini} and {\it G. Verzini} [Infinitely many solutions to fourth order superlinear periodic problems (preprint)] and by {\it M. Henrard} and {\it F. Sadyrbaev} [Nonlinear Anal., Theory Methods Appl. 33, 281-302 (1998; Zbl 0937.34020)].

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