Existence results on prescribing zero scalar curvature

Abstract

We study some time independent Schrödinger equations on complete noncompact manifolds. By showing that these equations have global positive solutions bounded away from zero, we prove that a large class of manifolds are conformal to complete manifolds with zero scalar curvature. They include those with Ricci curvature being nonnegative and the scalar curvature satisfying $V(x) \le C/d^{2}(x, x_{0})$ for an arbitrary $C>0$. As we have shown that there are noncompact manifolds with positive scalar curvature, which are not conformal to manifolds with positive constant scalar curvature, the issue of prescribing zero scalar curvature becomes interesting. The current result gives a partial answer to the Yamabe problem on noncompact manifolds with nonnegative Ricci curvatures.