Abstract
We study exact multiplicity of positive solutions and the bifurcation curve of the p-Laplacian perturbed Gelfand problem from combustion theory{((φ <sub>p</sub> (u <sup>′</sup> (x))) <sup>′</sup> + λ exp (frac(a u, a + u)) = 0, - 1 < x < 1,; u (- 1) = u (1) = 0,) where p > 1, φ <sub>p</sub> (y) = | y | <sup>p - 2</sup> y, (φ <sub>p</sub> (u <sup>′</sup> )) <sup>′</sup> 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 (frac(a u, a + u)) is the temperature dependence obeying the Arrhenius reaction-rate law. We find explicitly over(a, ̃) = over(a, ̃) (p) > 0 such that, if the activation energy a ≥ over(a, ̃), then the bifurcation curve is S-shaped in the (λ, {norm of matrix} u {norm of matrix} <sub>∞</sub> )-plane. More precisely, there exist 0 < λ <sub>*</sub> < λ <sup>*</sup> < ∞ such that the problem has exactly three positive solutions for λ <sub>*</sub> < λ < λ <sup>*</sup> , exactly two positive solutions for λ = λ <sub>*</sub> and λ = λ <sup>*</sup> , and a unique positive solution for 0 < λ < λ <sub>*</sub> and λ <sup>*</sup> < λ < ∞. © 2007 Elsevier Inc. All rights reserved.