Positive solutions to indefinite Neumann problems when the weight has positive average

TL;DR

This paper establishes the existence of positive solutions for a class of Neumann boundary value problems with indefinite weights that have a positive average, extending previous results to new weight conditions.

Contribution

It introduces new existence results for positive solutions when the weight has a positive mean, using a shooting method for a transformed planar system.

Findings

Existence of positive solutions under positive average weight conditions.

Application of a shooting method to a planar system.

Extension of previous results to weights with small positive mean.

Abstract

We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE $$ u" + q(t)g(u) = 0, \quad t \in [0, T], $$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(x) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$ x' = y, \qquad y' = h(x)y^2 + q(t), $$ with $h(x)$ a continuous function defined on the whole real line.