Exact multiplicity of positive solutions for concave–convex and convex–concave nonlinearities

Abstract

This note gives an unified treatment of the exact multiplicity results for both S-shaped and reversed S-shaped bifurcation for positive solutions of the two-point problemu″+λf(u)=0,for −1<x<1,u(−1)=u(1)=0, depending on a positive parameter λ, for both concave–convex and convex–concave nonlinearities f(u).