Rich dynamics in planar systems with heterogeneous nonnegative weights

TL;DR

This paper explores the complex structure of solutions for a generalized Sturm--Liouville problem with heterogeneous weights, extending classical models to include the full real parameter range and revealing rich dynamics.

Contribution

It provides the first comprehensive analysis of the global solution structure for the problem with non-negative weights and arbitrary real parameters, generalizing previous work.

Findings

Characterization of nodal solutions' structure

Extension to the full real line for parameter λ

Identification of rich solution dynamics

Abstract

This paper studies the global structure of the set of nodal solutions of a generalized Sturm--Liouville boundary value problem associated to the quasilinear equation $$ -(\phi(u'))'= \lambda u + a(t)g(u), \quad \lambda\in {\mathbb R}, $$ where $a(t)$ is non-negative with some positive humps separated away by intervals of degeneracy where $a\equiv 0$. When $\phi(s)=s$ this equation includes a generalized prototype of a classical model going back to Moore and Nehari, 1959. This is the first paper where the general case when $\lambda\in\mathbb{R}$ has been addressed when $a\gneq 0$. The semilinear case with $a\lneq 0$ has been recently treated by L\'{o}pez-G\'{o}mez and Rabinowitz.