Nodal solutions of weighted indefinite problems

TL;DR

This paper investigates the structure and bifurcation behavior of nodal solutions in one-dimensional indefinite boundary value problems with spectral parameters, combining analytical and numerical methods.

Contribution

It provides new insights into the bifurcation points and eigenvalue properties of indefinite problems, highlighting the interplay of analysis and numerics.

Findings

High order eigenvalues may not be concave.

Nodal solutions can bifurcate from multiple points.

Analytical and numerical methods complement each other.

Abstract

This paper analyzes the structure of the set of nodal solutions of a class of one-dimensional superlinear indefinite boundary values problems with an indefinite weight functions in front of the spectral parameter. Quite astonishingly, the associated high order eigenvalues might not be concave as it is the lowest one. As a consequence, in many circumstances the nodal solutions can bifurcate from three or even four bifurcation points from the trivial solution. This paper combines analytical and numerical tools. The analysis carried over on it is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously the numerical study confirms and illuminate the analysis.