Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $P$ be a selfadjoint, elliptic differential operator defined on $C^\infty(M)$ with smooth coefficients. Suppose as well that the lowest eigenvalue of $P$ is positive, i.e., $P$ is coercive. Is there a smooth function $u$ such that $u>0$ and $Pu>0$?
