On the number of positive solutions to an indefinite parameter-dependent Neumann problem

TL;DR

This paper analyzes how the number and structure of positive solutions to a parameter-dependent Neumann boundary value problem change with parameters, revealing three bifurcation scenarios and providing the first qualitative bifurcation diagram for such problems.

Contribution

It introduces a detailed topological and bifurcation analysis of positive solutions for an indefinite weight Neumann problem, including the first qualitative bifurcation diagram in the literature.

Findings

Identifies three distinct global bifurcation diagrams based on parameters.

Provides a phase-plane and time-mapping analysis for solution structure.

Depicts the first qualitative bifurcation diagram in the literature for this problem class.

Abstract

We study the second-order boundary value problem \begin{equation*} \begin{cases} \, -u''=a_{\lambda,\mu}(t) \, u^{2}(1-u), & t\in(0,1), \\ \, u'(0)=0, \quad u'(1)=0, \end{cases} \end{equation*} where $a_{\lambda,\mu}$ is a step-wise indefinite weight function, precisely $a_{\lambda,\mu}\equiv\lambda$ in $[0,\sigma]\cup[1-\sigma,1]$ and $a_{\lambda,\mu}\equiv-\mu$ in $(\sigma,1-\sigma)$, for some $\sigma\in\left(0,\frac{1}{2}\right)$, with $\lambda$ and $\mu$ positive real parameters. We investigate the topological structure of the set of positive solutions which lie in $(0,1)$ as $\lambda$ and $\mu$ vary. Depending on $\lambda$ and based on a phase-plane analysis and on time-mapping estimates, our findings lead to three different (from the topological point of view) global bifurcation diagrams of the solutions in terms of the parameter $\mu$. Finally, for the first time in the literature, a qualitative bifurcation diagram concerning the number of solutions in the $(\lambda,\mu)$-plane is depicted. The analyzed Neumann problem has an application in the analysis of stationary solutions to reaction-diffusion equations in population genetics driven by migration and selection.