High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions

TL;DR

This paper demonstrates that certain superlinear indefinite boundary value problems with Neumann conditions can have arbitrarily many positive solutions, depending on parameter choices, through phase-plane analysis and bifurcation techniques.

Contribution

It introduces a sharp phase-plane analysis for piecewise constant weights and explores how the number of solutions varies with the positive region's amplitude.

Findings

Existence of arbitrarily many positive solutions.

Dependence of solution count on the positive region's size.

Construction of global bifurcation diagrams.

Abstract

We prove that a class of superlinear indefinite problems with homogeneous Neumann boundary conditions admits an arbitrarily high number of positive solutions, provided that the parameters of the problem are adequately chosen. The sign-changing weight in front of the nonlinearity is taken to be piecewise constant, which allows to perform a sharp phase-plane analysis, firstly to study the sets of points reached at the end of the regions where the weight is negative, and then to connect such sets through the flow in the positive part. Moreover, we study how the number of solutions depends on the amplitude of the region in which the weight is positive, using the latter as the main bifurcation parameter and constructing the corresponding global bifurcation diagrams.