Exact multiplicity results for a p-Laplacian problem with concave–convex–concave nonlinearities

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 2( 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.