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

Abstract

This paper analyzes the structure of the set of positive solutions of \eqref{1.1}, where $a\equiv a_h$ is the piece-wise constant function defined in \eqref{1.3} for some $h\in (0,1)$. In our analysis, $\l$ 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 the numerical experiments of \cite{CLGT-2024}, we have analyzed the asymptotic behavior of the positive solutions of \eqref{1.1} as $h\uparrow 1$, when the shadow system of \eqref{1.1} 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 $\l<\pi^2$.