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

This paper analyzes the structure of the set of positive solutions of (1.1), where a≡ah is the piece-wise constant function defined in (1.3) for some h∈(0,1). In our analysis, λ 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 [2], we have analyzed the asymptotic behavior of the positive solutions of (1.1) as h↑1, when the shadow system of (1.1) is the linear equation −u″=π2u. 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↑1 if λ<π2.