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 ) = λ exp ( β u β + u ) , 0 < x < 1 , u ( 0 ) = 0 , u ( 1 ) u ( 1 ) + 1 u ′ ( 1 ) + [ 1 − u ( 1 ) u ( 1 ) + 1 ] u ( 1 ) = 0 , where λ > 0 is called the Frank–Kamenetskii parameter or ignition parameter, β > 0 is the activation energy parameter, u ( x ) is the dimensionless temperature, and the reaction term exp ( β u β + u ) is the temperature dependence obeying the simple Arrhenius reaction-rate law. We prove rigorously that, for β > β 1 ≈ 6.459 for some constant β 1 , the bifurcation curve is double S-shaped on the ( 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.].