Global structure of positive solutions for superlinear 2mth-boundary value problems

Abstract

We consider boundary value problems for nonlinear 2mth-order eigenvalue problem $$ \begin{gathered} ( - 1)^m u^{(2m)} (t) = \lambda a(t)f(u(t)),0 < t < 1, \hfill u^{(2i)} (0) = u^{(2i)} (1) = 0,i = 0,1,2,...,m - 1. \hfill \end{gathered} $$ . where a ∈ C([0, 1], [0, ∞)) and a(t0) > 0 for some t0 ∈ [0, 1], f ∈ C([0, ∞), [0, ∞)) and f(s) > 0 for s > 0, and f0 = ∞, where \( \mathop {\lim }\limits_{s \to 0^ + } f(s)/s \). We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.