I was wondering is there a sufficient condition (or sufficient and necessary condition) for the existence of positive solutions of the following linear PDE on a closed manifold $(M, g)$,
\begin{equation*} \Delta u +\nabla u\nabla f +hu=0. \end{equation*} where $f, h\in C^{\infty}(M)$.
I got some necessary conditions using the Stokes formula, but I couldn't find a statement for sufficient condition, or sufficient and necessary condition. Thank you very much for any suggestions.
Not a full answer, but some observations.
First, I'm assuming $M$ is compact, orientable, etc. $h$ attains its maximum $C$ somewhere on $M$, so that $k(q) = C - h(q)$ is nonnegative.
Your PDE then becomes the eigenvalue problem $$Lu = Cu$$ where $$L = -e^{-f}\nabla \cdot \left(e^f \nabla u\right) + k u$$ is a positive elliptic operator on $M$. For given $f$ and $k$, this equation will have nontrivial solutions only for countably many positive values of $C$, and nonnegative solutions $u$ if and only if $C$ is the least eigenvalue of $L$.
I have no idea how to go about formulating a sufficient condition for $C$ being that eigenvalue, though.