Global behavior of the branch of positive solutions for nonlinear Sturm-Liouville problems (Q997542)

The author considers the following nonlinear Sturm-Liouville problem: \[ \begin{cases} -u^{\prime\prime}(t) +f(u(t))=\lambda u(t),\quad t \in I:=(0, 1) \\ u(t)>0, \quad t\in I, \\ u(0)=u(1)=0, \end{cases} \] where \(\lambda >0\) is an eigenvalue parameter. Under suitable assumptions on \(f\), it is known that for every \(\alpha>0\) there exists a unique solution \((\lambda,u)=(\lambda(\alpha),u_{\alpha})\in I\!\!R_+ \times C^2(\overline{I})\), with \(\| u_{\alpha}\|=\alpha\). The aim of this work is to find a precise asymptotic formula for \(\lambda(\alpha)\) when \(\alpha \to \infty\). It is proved that, if for some \(p>1\) the function \(h(u):= \frac{f(u)}{u^p}\), for \(u\geq0\), satisfies adequate conditions then \(\lambda(\alpha) \sim \alpha^{p-1}h(\alpha)\) as \(\alpha \to \infty\).