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Zbl 1136.34016
Xu, Jia; Han, XiaoLing
Existence of nodal solutions for Lidstone eigenvalue problems.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 67, No. 12, A, 3350-3356 (2007). ISSN 0362-546X

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.
[Sergei A. Brykalov (Ekaterinburg)]
MSC 2000:
*34B15 Nonlinear boundary value problems of ODE

Keywords: Lidstone eigenvalue problem; Nodal solution; Existence; Bifurcation methods

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