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

Abstract

We study the exact multiplicity and ordering properties of positive solutions of the p-Laplacian Dirichlet problem − ϕ p u′(x) ′=λf(u), −1<x<1,u(−1)=u(1)=0, where p>1, ϕ p ( y)=| y| p−2 y, ( ϕ p ( u′))′ is the one-dimensional p-Laplacian, and λ>0 is a bifurcation parameter. Assuming that f∈ C[0,∞)∩ C 2(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.