Qualitative behavior and computation of multiple solutions of singular nonlinear boundary value problems

Abstract

We consider boundary value problems of the form

$$y^{\prime \prime }=-f\left(t,y\right),y\left(0\right)=0,y\left(1\right)=0,$$

motivated by examples where $f\left(t,y\right)=\varphi \left(t\right)g\left(y\right)$ and $g\left(y\right)$ behave like $y^{-\lambda }$ ($\lambda >0$) as $y\to 0^{+}$. We explore conditions under which such problems have multiple positive solutions, investigate qualitative behavior of these solutions, and discuss computational methods for approximating the solutions.