Existence and nonuniqueness of solutions to a functional-differential equation

We examine the functional-differential equation $\Delta u(\boldsymbol x)-\operatorname{div}(u(H(\boldsymbol x))\mathbf f(\boldsymbol x))=0$ on a torus which is a generalization of the stationary Fokker–Planck equation. Under sufficiently general assumptions on the vector field $\mathbf f$ and the map $H$, we prove the existence of a nontrivial solution. In some cases the subspace of solutions is established to be multidimensional.