Exact multiplicity results for a class of boundary-value problems with cubic nonlinearities (Q1900425)

The paper deals with multiplicity results for the nonlinear Dirichlet boundary value problem \((*)\) \(u'' + \lambda u (u - a(x)) (b(x) - u) = 0\) \(u (-1) = 0 = u(1)\). Conditions on the functions \(a,b\) are given which guarantee that only the two following possibilities can occur: (i) BVP \((*)\) has no solution for any \(\lambda > 0\). (ii) There exists \(\lambda_0\) such that \((*)\) has either zero, one, or two solutions depending on whether \(\lambda < \lambda_0\), \(\lambda = \lambda_0\), or \(\lambda > \lambda_0\), respectively. The principal method used in the proof of this statement is the Crandall-Rabinowitz bifurcation theorem.