In this interesting manuscript, the authors investigate the existence of positive solutions of the second order nonlinear ordinary differential equation \[ u''+q(t)g(u)=0,\tag{1} \] where the weight \(q\) is allowed to change sign and \(g\) is superlinear at \(0\), sublinear at infinity and positive on \((0, +\infty)\), under Dirichlet, Neumann and periodic boundary conditions. A remarkable feature here is that the authors, roughly speaking, “play” with the nodal behaviour of the weight function in order to achieve multiple positive solutions. The proofs utilize, in a careful way, topological degree arguments and in particular Mawhin’s coincidence degree for the Neumann and periodic cases. Furthermore, the authors provide general properties of the positive solutions of (1) on the whole real line, study positive subharmonic solutions and solutions with a chaotic-like behaviour. Finally, the authors give an application to radial solutions of PDEs on annular domains under Dirichlet or Neumann boundary conditions.