S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws

We study the bifurcation curve and exact multiplicity of positive solutionsof the combustion problem with general Arrhenius reaction-rate laws\begin{eqnarray}u^{\prime \prime }(x)+\lambda (1+\epsilon u)^{m}e^{\frac{u}{1+\epsilon u}}=0, -1 0$ and $-\infty < m <1$. We prove that, for $(-4.103\approx)$ $\tilde{m}\leq m < 1$ for someconstant $\tilde{m}$, the bifurcation curve is S-shaped on the $(\lambda, \|u\|_{\infty })$-plane if $0<\epsilon \leq \frac{6}{7}\epsilon _{\text{tr}}^{\text{Sem}}(m)$, where\begin{eqnarray}\epsilon _{\text{tr}}^{\text{Sem}}(m)=\left\{\begin{array}{l}(\frac{1-\sqrt{1-m}}{m})^{2}\ \text{ for }-\infty < m < 1, m \neq 0, \\\frac{1}{4}\ \text{for}\ m=0,\end{array}\right.\end{eqnarray}is the Semenov transitional value for general Arrhenius kinetics. Inaddition, for $-\infty < m < 1$, the bifurcation curve is S-like shaped if $0<\epsilon \leq \frac{8}{9} \epsilon _{\text{tr}}^{\text{Sem}}(m).$ Our results improve and extend those in Wang (Proc. Roy. Soc. London Sect. A, 454 (1998), 1031--1048.)