High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type

Resumen

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in Moore and Nehari (1959). Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number kappa >= 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 down arrow -infinity, 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 high multiplicity also holds true for a much larger class of weights, even arbitrarily close to situations where there is uniqueness of positive solutions.