  A variant of Li-Tam theory, which associates to each end of a complete
Riemannian manifold a positive solution of a given Schr\"odinger equation on
the manifold, is developed. It is demonstrated that such positive solutions
must be of polynomial growth of fixed order under a suitable scaling invariant
Sobolev inequality. Consequently, a finiteness result for the number of ends
follows. In the case when the Sobolev inequality is of particular type, the
finiteness result is proven directly. As an application, an estimate on the
number of ends for shrinking gradient Ricci solitons and submanifolds of
Euclidean space is obtained.
