Uniqueness, non-degeneracy, and exact multiplicity of positive solutions for superlinear elliptic problems

TL;DR

This paper investigates positive solutions to certain superlinear elliptic problems, establishing conditions for their uniqueness, non-degeneracy, and exact multiplicity using shooting methods, and extends previous multiplicity results with numerical examples and open problems.

Contribution

It provides new non-degeneracy and multiplicity results for positive solutions of nonlinear differential equations with sign-changing weights, extending prior work by Feltrin and Zanolin.

Findings

Established conditions for uniqueness and multiplicity of positive solutions.

Demonstrated non-degeneracy of solutions under certain conditions.

Presented numerical examples and discussed open problems.

Abstract

In this paper, we focus our attention on the positive solutions to second-order nonlinear ordinary differential equations of the form $u''+q(t)g(u)=0$, where $q$ is a sign-changing weight and $g$ is a superlinear function. We exploit the classical shooting approach and the comparison theorem to present non-degeneracy and exact multiplicity results for positive solutions. This completes the multiplicity results obtained by Feltrin and Zanolin. Numerical examples and some related open problems are also discussed.