Article

Czechoslovak Mathematical Journal

, Volume 56, Issue 4, pp 1243-1263

First online:

Nodal solutions for a second-order m-point boundary value problem

  • Ruyun MaAffiliated withDepartment of Mathematics, Northwest Normal University

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Abstract

We study the existence of nodal solutions of the m-point boundary value problem
$$\begin{gathered} u'' + f(u) = 0,0 < t < 1, \hfill \\ u'(0) = 0,u(1) = \sum\limits_{i = 1}^{m - 2} {\alpha _i u(\eta i)} \hfill \\ \end{gathered} $$
where η i ∈ ℚ (i = 1, 2, ..., m − 2) with 0 < η 1 < η 2 < ... < η m−2 < 1, and α i ∈ ℝ (i = 1, 2, ..., m − 2) with α i > 0 and \(\sum\nolimits_{i = 1}^{m - 2} {\alpha _i } \) < 1. We give conditions on the ratio f(s)/s at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.

Keywords

multiplicity results eigenvalues bifurcation methods nodal zeros multi-point boundary value problems