Exact multiplicity and ordering properties of positive solutions of a p-Laplacian Dirichlet problem and their applications ✩

We study the exact multiplicity and ordering properties of positive solutions of the p-Laplacian Dirichlet problem − ϕp u (x) = λf (u), −1 1, ϕp(y) = |y|p−2y, (ϕp(u )) is the one-dimensional p-Laplacian, and λ > 0 is a bifurcation parameter. Assuming that f ∈ C[0,∞) ∩C2(0,∞) satisfies (F1)–(F4), we show that the bifurcation curve has exactly one critical point, a maximum, on the ( u ∞,λ)-plane. Thus we are able to determine the exact multiplicity of positive solutions. We give two interesting applications for a nonlinear Dirichlet problem of polynomial nonlinearities with positive coefficients and for a stationary singular diffusion problem.  2003 Elsevier Inc. All rights reserved