Abstract
We study the exact multiplicity and ordering properties of positive solutions of the p-Laplacian Dirichlet problem -( <sub>p</sub> (u′ (x)))′ = λf (u), -1 < × < 1, u(-1) = u(1) = 0, where p > 1, <sub>p</sub> (y) = y <sup>p-2</sup> y, ( <sub>p</sub> (u′))′ is the one-dimensional p-Laplacian, and λ > 0 is a bifurcation parameter. Assuming that f ∈ C[0, ∞ ∩ C <sup>2</sup> (0, ∞) satisfies (F1)-(F4), we show that the bifurcation curve has exactly one critical point, a maximum, on the (∥ u ∥ <sub>∞</sub> , 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.