Splitting and gap theorems in the presence of a Poincaré-Sobolev inequality

Abstract

Up to now, we have been using Theorem 4.5 to show that solutions of a differential problem of the type

$$ \left\{ \begin{gathered} \psi \Delta \psi + a\left( x \right)\psi ^2 \geqslant - A\left| {\left. {\nabla \psi } \right|^2 ,} \right. \hfill \\ \psi \geqslant 0 \hfill \\ \end{gathered} \right. $$

have to be identically zero. The aim of this section is to present a geometrical problem in which the second alternative of Theorem 4.5 does actually occur, that is, ψ becomes a positive solution of the linear equation

$$ \Delta \psi + a\left( x \right)\psi = 0 $$

.