Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line

TL;DR

This paper proves the existence of infinitely many positive solutions with complex behaviors for a superlinear indefinite ODE on the real line, using variational methods and Nehari constraints.

Contribution

It introduces a novel variational approach to establish infinitely many positive solutions with complex patterns for superlinear indefinite ODEs.

Findings

Existence of infinitely many positive solutions with complex behavior.

Construction of solutions with arbitrarily large minimal periods.

Identification of bounded non-periodic solutions exhibiting complex dynamics.

Abstract

We show the existence of infinitely many positive solutions, defined on the real line, for the nonlinear scalar ODE \[ \ddot u + (a^+(t) - \mu a^-(t)) u^3 = 0, \] where $a$ is a periodic, sign-changing function, and the parameter $\mu>0$ is large. Such solutions are characterized by the fact of being either small or large in each interval of positivity of $a$. In this way, we find periodic solutions, having minimal period arbitrarily large, and bounded non-periodic solutions, exhibiting a complex behavior. The proof is variational, exploiting suitable natural constraints of Nehari type.