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)))′+λexp(aua+u)=0,−11p>1, φp(y)=|y|p−2yφp(y)=|y|p−2y, (φp(u′))′(φp(u′))′ is the one-dimensional p  -Laplacian, λ>0λ>0 is the Frank–Kamenetskii parameter, u(x)u(x) is the dimensionless temperature, and the reaction term f(u)=exp(aua+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‖∞)(λ,‖u‖∞)-plane. More precisely, there exist 0<λ*<λ*<∞0<λ*<λ*<∞ such that the problem has exactly three positive solutions for λ*<λ<λ*λ*<λ<λ*, exactly two positive solutions for λ=λ*λ=λ* and λ=λ*λ=λ*, and a unique positive solution for 0<λ<λ*0<λ<λ* and λ*<λ<∞λ*<λ<∞.