  In this paper we consider a superlinear one-dimensional elliptic boundary
value problem that generalizes the one studied by Moore and Nehari in [43].
Specifically, we deal with piecewise-constant weight functions in front of the
nonlinearity with an arbitrary number $\kappa\geq 1$ of vanishing regions. We
study, from an analytic and numerical point of view, the number of positive
solutions, depending on the value of a parameter $\lambda$ and on $\kappa$.
  Our main results are twofold. On the one hand, we study analytically the
behavior of the solutions, as $\lambda\downarrow-\infty$, in the regions where
the weight vanishes. Our result leads us to conjecture the existence of
$2^{\kappa+1}-1$ solutions for sufficiently negative $\lambda$. On the other
hand, we support such a conjecture with the results of numerical simulations
which also shed light on the structure of the global bifurcation diagrams in
$\lambda$ and the profiles of positive solutions.
  Finally, we give additional numerical results suggesting that the same high
multiplicity result holds true for a much larger class of weights, also
arbitrarily close to situations where there is uniqueness of positive
solutions.
