Let $(M,g)$ be a smooth, closed Riemannian manifold and let $X$ and $c$ respectively be a smooth vector field and a smooth function on $M$. Do there exist general criteria for determining for which smooth functions $f$ there exists a smooth solution to the elliptic PDE $$\Delta u +X(u) +cu =f?$$ I'm familiar with the local theory which guarantees for us the existence of local solutions in any coordinate patch but there seem to be global obstructions once we move to the manifold setting. For example, in the instance that $X$ and $c$ are both identically $0$, there is a clear global obstruction to the problem, namely whether or not the integral of $f$ over $M$ vanishes (one can show, using the heat flow [or sheaf cohomology in the instance that $M$ is a Kähler] that, in this instance, this is in fact the only obstruction). I believe that in the case that either $c \geq 0$ or $c \leq 0$ the local theory combined with sheaf cohomology tells us that the answer is essentially the same as for the Laplacian, but if we don't have control over $c$, then what happens?
Any and all insights are welcomed!