# Positive subharmonic solutions to superlinear ODEs with indefinite weight

## TL;DR

This paper proves the existence of positive subharmonic solutions of any order for a class of superlinear second-order ODEs with indefinite periodic weights, using topological and degree theory methods.

## Contribution

It establishes new existence results for positive subharmonic solutions under sharp mean value conditions and large negative weight parts, combining Mawhin's degree theory with Poincaré-Birkhoff theorem.

## Key findings

- Existence of positive subharmonic solutions of any order under mean value condition.
- Existence of positive subharmonics for large negative parts of the weight.
- Application of coincidence degree theory and Poincaré-Birkhoff theorem.

## Abstract

We study the positive subharmonic solutions to the second order nonlinear ordinary differential equation \begin{equation*} u'' + q(t) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth both at zero and at infinity, and $q(t)$ is a $T$-periodic sign-changing weight. Under the sharp mean value condition $\int_{0}^{T} q(t) ~\!dt < 0$, combining Mawhin's coincidence degree theory with the Poincar\'e-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order $k$ for any large integer $k$. Moreover, when the negative part of $q(t)$ is sufficiently large, using a topological approach still based on coincidence degree theory, we obtain the existence of positive subharmonics of order $k$ for any integer $k\geq2$.