Global structure of positive solutions for superlinear second order $m$-point boundary value problems

Abstract

In this paper, we consider the nonlinear eigenvalue problems \begin{gather*} u''+\lambda h(t)f(u)=0, \quad 0< t< 1, \\ u(0)=0,\quad u(1)=\sum_{i=1}^{m-2}\alpha_i u(\eta_i), \end{gather*} where $m\geq 3$, $ \eta_i\in (0,1)$ and $\alpha_i> 0$ for $i=1,\ldots,m-2$, with $\sum_{i=1}^{m-2}\alpha_i\eta_i< 1$; $h\in C([0,1], [0,\infty))$ and $h(t)\ge 0$ for $t\in [0,1]$ and $h(t_0)> 0$ for $t_0\in [0,1]$; $f\in C([0,\infty),[0,\infty))$ and $f(s)> 0$ for $s> 0$, and $f_0=\infty$, where $f_0=\lim_{s\rightarrow 0^+}f(s)/s$. We investigate the global structure of positive solutions by using the nonlinear Krein-Rutman Theorem.