Existence of nodal solutions for Lidstone eigenvalue problems. (English) Zbl 1136.34016

Xu, Jia; Han, XiaoLing

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 12, 3350-3356 (2007).

The authors study nodal solutions to the Lidstone eigenvalue problem \[ (-1)^m u^{(2m)}(x)= ra(x) f(u(x)),\qquad 0< x< 1, \]
\[ u^{(2i)}(0)= u^{(2i)}(1)= 0,\qquad i= 0,\dots, m-1. \] Sufficient conditions are obtained for the existence of solutions with exactly \(k-1\) simple zeros in \((0,1)\). These conditions use the eigenvalue \(\lambda_k\) of the corresponding linear problem. Illustrative examples are given.

Reviewer: Sergei A. Brykalov (Ekaterinburg)

MSC:

34B15

Nonlinear boundary value problems for ordinary differential equations

Keywords:

Lidstone eigenvalue problem; Nodal solution; Existence; Bifurcation methods

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