Consider the elliptic operator $Lu = – Delta u + langle nabla u , X rangle + c , u $ acting on functions on a closed Riemannian manifold $M$. Here $Delta$ denotes the Laplace-Beltrami operator, $X$ is an arbitrary smooth vector field, and $c geq 0$ is a smooth function on $M$ which does not vanish identically. Does $L$ have a so-called `principal eigenvalue’ $lambda_1 > 0$, whose corresponding (unique up to scaling) eigenfunction does not change sign?

A similar statement holds for smooth domains in $mathbb{R}^n$, as shown for instance in Evans’ PDE book, chapter 6. Moreover, in this paper it is sated that this fact is equivalent to the operator satisfying a maximum principle (which is indeed the case for the above $L$).