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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

$Lu\left(x\right)=p\left(x\right)u\left(x\right)+g\left(x,{u}^{\left(0\right)}\left(x\right),\cdots ,{u}^{\left(2m-1\right)}\left(x\right)\right)u\left(x\right),$

where $L$ is a selfadjoint, disconjugate operator on $\left[0,\pi \right]$ and the boundary conditions are separated. It is assumed that $g$ is “superlinear at infinity” and that ${lim}_{|\xi |\to 0}g\left(x,\xi \right)=0$. 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 $Lu=\mu pu$.

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)].

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 47J15 Abstract bifurcation theory 34C23 Bifurcation (ODE)