Existence of nodal solutions of nonlinear Lidstone boundary value problems

Abstract

We investigate the existence of nodal solutions for the nonlinear Lidstone boundary value problem \begin{document}$ \begin{align} \left\{\begin{array}{ll} (-1)^m (u^{(2m)}(t)+c u^{(2m-2)}(t)) = \lambda a(t)f(u), \; \; \ \ \ t\in (0, r), \\ u^{(2i)}(0) = u^{(2i)}(r) = 0, \ \ i = 0, 1, \cdots, m-1, \end{array} \right.~~(P) \end{align} $\end{document} where $ \lambda > 0 $ is a parameter, $ c $ is a constant, $ m\geq1 $ is an integer, $ a :[0, r]\rightarrow [0, \infty) $ is continuous with $ a\not\equiv0 $ on the subinterval within $ [0, r] $, and $ f: \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function. We analyze the spectrum structure of the corresponding linear eigenvalue problem via the disconjugacy theory and Elias's spectrum theory. As applications of our spectrum results, we show that problem $ (P) $ has nodal solutions under some suitable conditions. The bifurcation technique is used to obtain the main results of this paper.

Keywords

• lidstone
• nodal solutions
• spectrum
• bifurcation
• disconjugacy theory