Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree

TL;DR

This paper proves the existence of multiple positive periodic solutions for a superlinear indefinite differential equation with sign-changing weight, using topological methods and coincidence degree theory, applicable to various boundary conditions.

Contribution

It extends Mawhin's coincidence degree to open unbounded sets and demonstrates the multiplicity of solutions in superlinear indefinite problems with sign-changing weights.

Findings

Existence of 2^m - 1 positive solutions for large μ

Applicable to Neumann boundary conditions

Provides a topological approach to subharmonic solutions

Abstract

We study the periodic boundary value problem associated with the second order nonlinear differential equation $$ u" + c u' + \left(a^{+}(t) - \mu \, a^{-}(t)\right) g(u) = 0, $$ where $g(u)$ has superlinear growth at zero and at infinity, $a(t)$ is a periodic sign-changing weight, $c\in\mathbb{R}$ and $\mu>0$ is a real parameter. We prove the existence of $2^{m}-1$ positive solutions when $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a periodicity interval) and $\mu$ is sufficiently large. The proof is based on the extension of Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.