Three positive solutions to an indefinite Neumann problem: a shooting method (Q1683746)

The authors deal with the Neumann boundary value problem \[ \begin{aligned} u''+(\lambda a^+(t)-\mu a^-(t))g(u)&=0, \\ u'(0)&=u'(1)=0, \end{aligned} \] where the weight term has two positive humps separated by a negative one and \(g : [0, 1] \to R\) is a continuous function such that \(g(0) = g(1) = 0\), \(g(s) > 0\) for \(0 < s < 1\) and \(\lim_{s\to 0^+} g(s)/s = 0\). They prove the existence of three solutions when \(\lambda\) and \(\mu\) are positive and sufficiently large.