High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type
<|reference_start|>High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type: 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.<|reference_end|>
@article{cubillos2024high,
    title={High multiplicity of positive solutions in a superlinear problem of
  Moore-Nehari type},
    author={Pablo Cubillos, Juli'an L'opez-G'omez, Andrea Tellini},
    journal={arXiv preprint arXiv:2402.19084},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.19084},
    primaryClass={math.AP cs.NA math.CA math.NA}
}