Summary: We prove a necessary and a sufficient condition for the existence of a positive solution of the equation \((P-\mu W)u=0\) in \(\Omega\), where \(P\) is a critical, second-order, linear elliptic operator which is defined on a subdomain \(\Omega\) of a noncompact Riemannian manifold \(X\). It is assumed that \(W\in C^\alpha(\Omega)\) is a ``weak'' perturbation and \(\mu<0\) is small enough.