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

• Published in: Communications in nonlinear science & numerical simulation 2024-09, Vol.136, p.108118, Article 108118
• Main Authors: Cubillos, Pablo, López-Gómez, Julián, Tellini, Andrea
• Format: Article
• Language: English
• Subjects: Global bifurcation diagrams; Multiplicity of positive solutions; Numerical path-following; Superlinear problem
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2024.108118

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 κ≥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 λ and on κ. Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as λ↓−∞, in the regions where the weight vanishes. Our result leads us to conjecture the existence of 2κ+1−1 solutions for sufficiently negative λ. 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 λ 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. •We consider a heterogeneous superlinear one-dimensional BVP of Moore-Nehari type.•We study the behavior of positive solutions in the vanishing regions of the weight.•We conjecture the number of positive solutions based on that of vanishing regions.•We support this conjecture with numerical simulations of the bifurcation diagrams.•Simulations for positive weights show in addition that multiplicity is robust.