Existence of a double S-shaped bifurcation curve with six positive solutions for a combustion problem

We study the bifurcation curve of positive solutions of the combustion problem with nonlinear boundary conditions given by{−u″(x)=λexp(βuβ+u),00λ>0 is called the Frank–Kamenetskii parameter or ignition parameter, β>0β>0 is the activation energy parameter, u(x)u(x) is the dimensionless temperature, and the reaction term exp(βuβ+u) is the temperature dependence obeying the simple Arrhenius reaction-rate law. We prove rigorously that, for β>β1≈6.459β>β1≈6.459 for some constant β1β1, the bifurcation curve is double S-shaped on the (λ,‖u‖∞)(λ,‖u‖∞)-plane and the problem has at least six positive solutions for a certain range of positive λ. We give rigorous proofs of some computational results of Goddard II, Shivaji and Lee [J. Goddard II, R. Shivaji, E.K. Lee, A double S-shaped bifurcation curve for a reaction–diffusion model with nonlinear boundary conditions, Bound. Value Probl. (2010), Art. ID 357542, 23 pp.].