Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities (Q743876)

The authors give an unified treatment of the exact multiplicity results for both \(S\)-shaped and reversed \(S\)-shaped bifurcation for positive solutions of the two-point problem \[ u''+\lambda f(u)=0, \quad u(-1)=u(1)=0, \] depending on a positive parameter \(\lambda\), for both concave-convex and convex-concave nonlinearities \(f\).