Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems

TL;DR

This paper establishes conditions for positive solutions of second-order nonlinear equations with sign-changing weights using Mawhin's coincidence degree, applicable to periodic and Neumann boundary value problems, with extensions to PDEs in annular domains.

Contribution

It introduces new existence criteria for positive solutions in superlinear cases using coincidence degree, covering both periodic and Neumann boundary conditions.

Findings

Existence of positive solutions under superlinear growth conditions.

Applicable to equations with sign-changing weight functions.

Extended to nonlinear PDEs in annular domains.

Abstract

We prove the existence of positive periodic solutions for the second order nonlinear equation $u" + a(x) g(u) = 0$, where $g(u)$ has superlinear growth at zero and at infinity. The weight function $a(x)$ is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.