Exact multiplicity of solutions and S-shaped bifurcation curves for the p-Laplacian perturbed Gelfand problem in one space variable ✩

We study exact multiplicity of positive solutions and the bifurcation curve of the p-Laplacian perturbed Gelfand problem from combustion theory ⎧⎨⎩ ϕp u (x) + λ exp au a + u = 0, −11, ϕp(y) = |y|p−2y, (ϕp(u )) is the one-dimensional p-Laplacian, λ>0 is the Frank–Kamenetskii parameter, u(x) is the dimensionless temperature, and the reaction term f (u) = exp( au a+u ) is the temperature dependence obeying the Arrhenius reaction-rate law. We find explicitly ˜a = ˜a(p) > 0 such that, if the activation energy a ˜a, then the bifurcation curve is S-shaped in the (λ, u ∞)-plane. More precisely, there exist 0<λ∗ <λ∗ <∞such that the problem has exactly three positive solutions for λ∗ <λ<λ∗, exactly two positive solutions for λ = λ∗ and λ = λ∗, and a unique positive solution for 0<λ<λ∗ and λ∗ <λ<∞. © 2007 Elsevier Inc. All rights reserved