Abstract
We study the bifurcation curve and exact multiplicity of positive solutions of the combustion problem with general Arrhenius reaction-rate laws { u"(x) + λ(1 + εu)m <sub>e</sub> u/1+εu = 0, - 1 < x < 1, u(-1) = u(1) = 0, where the bifurcation parameters λ ε < 0 and -∞ < m < 1. We prove that, for (-4.103) ̃m ≤ m < 1 for some constant ̃m, the bifurcation curve is S-shaped on the (λ, u∞)-plane if 0 < ε ≤ 6/7 εSemεtr (m), where εSem tr (m) = { ( 1 <sup>1-m</sup> /m ) <sup>2</sup> for -∞ < m < 1; m = 0; 1/4 for m = 0; is the Semenov transitional value for general Arrhenius kinetics. In addition, for -1 < m < 1, the bifurcation curve is S-like shaped if 0 < ε < 8/9 εSem tr (m). Our results improve and extend those in Wang (Proc. Roy. Soc. London Sect. A, 454 (1998), 1031{1048.).