Global bifurcation diagrams of positive solutions for a class of 1-D superlinear indefinite problems

TL;DR

This paper combines analytical and numerical methods to explore the structure of positive solutions in one-dimensional superlinear indefinite boundary value problems, confirming conjectures and revealing complex solution behaviors.

Contribution

It provides new analytical results on solution decay and confirms a conjecture, while numerically illustrating the global solution structure for complex problem prototypes.

Findings

Fast decay of solutions as λ→-∞ in negative regions

Numerical confirmation of intricate solution structures

Validation of a conjecture under dynamical conditions

Abstract

This paper analyzes the structure of the set of positive solutions of a class of one-dimensional superlinear indefinite bvp's. It is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously its numerical study confirms and illuminates the analysis. On the analytical side, we establish the fast decay of the positive solutions as $\lambda\downarrow -\infty$ in the region where $a(x)<0$ (see (1.1)), as well as the decay of the solutions of the parabolic counterpart of the model (see (1.2)) as $\lambda\downarrow-\infty$ on any subinterval of $[0,1]$ where $u_0=0$, provided $u_0$ is a subsolution of (1.1). This result provides us with a proof of a conjecture of [28] under an additional condition of a dynamical nature. On the numerical side, this paper ascertains the global structure of the set of positive solutions on some paradigmatic prototypes whose intricate behavior is far from predictable from existing analytical results.