Existence of positive solutions of a superlinear boundary value problem with indefinite weight

TL;DR

This paper proves the existence of positive solutions for a nonlinear second-order boundary value problem with an indefinite weight function, using topological methods to extend previous results in the field.

Contribution

It introduces a new existence theorem for positive solutions of a superlinear boundary value problem with sign-changing weight, broadening the scope of prior work.

Findings

Established existence of at least one positive solution

Extended applicability to weights that change sign

Utilized Leray-Schauder degree for topological analysis

Abstract

We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change its sign. We assume that the function $g\colon\mathopen{[}0,+\infty\mathclose{[}\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, so as the case $g(s)=s^{p}$, with $p>1$, is covered. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.