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

In this paper, we focus our attention on the positive solutions to second-order nonlinear ordinary differential equations of the form $u^{\mathrm{\prime }\mathrm{\prime }}+q(t)g(u)=0$, 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.