Yamabe problem ensures that for any Riemannian metric $g$, in its conformal class $[g]$ there always exists a metric $\bar g$ whose scalar curvature $\bar R$ is constant.
I was wondering whether this is a possible way to find metrics with positive scalar curvature? i.e., for a Riemannian metric $g$, in its conformal class $[g]$ does there exist a metric $\bar g$ whose scalar curvature $\bar R$ is positive? If $R$ is positive, this trivial. If $R$ is negative, this is not possible by the conformal change of scalar curvature. If $R$ is positive somewhere and negative somewhere, is it possible to find a solution?
In other words, suppose $\phi>0$, is there a sufficient condition for $f$ such that $$-\Delta u+fu=\phi$$ has a positive solution?