A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem

Abstract

We study the global bifurcation and exact multiplicity of positive solutions for \begin{equation*} \begin{cases} u^{\prime \prime }(x)+\lambda f_{\varepsilon }(u)=0\text{,}\; \;-10$ is a bifurcation parameter, $\varepsilon \in \Theta $ is an evolution parameter, and $\Theta \equiv \left( \sigma _{1},\sigma_{2}\right) $ is an open interval with $0\leq \sigma _{1}<\sigma _{2}\leq \infty $. Under some suitable hypotheses on $f_{\varepsilon }$, we prove that there exists $\varepsilon _{0}\in \Theta $ such that, on the $(\lambda,\|u\|_{\infty })$-plane, the bifurcation curve is S-shaped for $\sigma_{1}<\varepsilon <\varepsilon _{0}$ and is monotone increasing for $\varepsilon _{0}\leq \varepsilon <\sigma _{2}$. We give an application to prove global bifurcation of bifurcation curves for the one-dimensional perturbed Gelfand problem.

Keywords

• global bifurcation
• multiparameter problem
• s-shaped bifurcation curve
• exact multiplicity
• positive solution