Abstract
We study the exact number of positive solutions of a two-point Dirichlet boundary-value problem involving the p-Laplacian operator. We consider the case p=2 as well as the case p > 1, when the nonlinearity f satisfies f(0)=0 and has two distinct simple positive zeros and such that f″ changes sign exactly twice on (0,∞). Note that we may allow that f″ changes sign more than twice on (0,∞). Some interesting examples of quartic polynomials are given. In particular, for f(u)=-u <sup>2</sup> (u-1)(u-2), we study the evolution of the bifurcation curves of the p-Laplacian problem as p increases from 1 to infinity, and hence are able to determine the exact multiplicity of positive solutions for each p > 1. © 2003 Elsevier Science Ltd. All rights reserved.