Global multiplicity results in a Moore-Nehari type problem with a spectral parameter

TL;DR

This paper investigates the multiplicity and asymptotic behavior of positive solutions in a Moore-Nehari problem with a spectral parameter, revealing new insights into solution structure as a deformation parameter approaches a critical value.

Contribution

It provides the first analysis of the problem's asymptotic behavior as the deformation parameter approaches 1, extending previous numerical results and addressing a singular perturbation challenge.

Findings

Established multiplicity results for positive solutions

Analyzed asymptotic behavior as h approaches 1

Identified blow-up behavior of solutions for certain λ

Abstract

This paper analyzes the structure of the set of positive solutions of a Moore-Nehari type problem, where $a\equiv a_h$ is a piece-wise constant function defined for some $h\in (0,1)$. In our analysis, $\lambda$ is regarded as a bifurcation parameter, whereas $h$ is viewed as a deformation parameter between the autonomous case when $a=1$ and the linear case when $a=0$. In this paper, besides establishing some of the multiplicity results suggested by previous numerical experiments (see Cubillos, L\'opez-G\'omez and Tellini, 2024), we have analyzed the asymptotic behavior of the positive solutions of the problem as $h\uparrow 1$, when the shadow system of the problem is the linear equation $-u''=\pi^2 u$. This is the first paper where such a problem has been addressed. Numerics is of no help in analyzing this singular perturbation problem because the positive solutions blow-up point-wise in $(0,1)$ as $h\uparrow 1$ if $\lambda<\pi^2$.