Abstract
We study global bifurcation curves and the exact multiplicity of positive solutions for the two-point boundary value problem arising in combustion theory:(Formula presented.)where (Formula presented.) is the Frank–Kamenetskii parameter and a > 0 is the activation energy parameter. We prove that there exists a critical bifurcation value a <sub>0</sub> (Formula presented.) such that, on the (Formula presented.)-plane, the bifurcation curve is S-shaped for (Formula presented.) and is monotone increasing for (Formula presented.). That is, we prove the long-standing conjecture for the one-dimensional perturbed Gelfand problem. We also study, in the (Formula presented.)-space, the shape and structure of the bifurcation surface.