Consider a closed Riemannian manifold $(M,g)$ and let $u \in C^{2,\alpha}(M)$ be a positive function on $M$.

I am interested on the existence of solution for the following problem: given a continuous function $\psi$ on $M$, when does there exists unique $v \in C^{2,\alpha}(M)$ such that $$\Delta v + F(u,\nabla u)v + G(u,\nabla u)g(\nabla u,\nabla v) = \psi$$ where $F$ and $G$ are smooth functions on their parameters and $u\in C^{2,\alpha}(M)$ is fixed.

This should not be a very difficult problem for one reason: this PDE is nothing more than a linear elliptic second order PDE, so it has a well consolidated theory.

The problem is that in general, to find a solution for this problem one needs to ensure that $\psi$ is in the orthogonal complement of the dual of the linear and elliptic operator $$P_u : v \mapsto \left(\Delta(\cdot) + F(u,\nabla u)\cdot + G(u,\nabla u)g(\nabla u,\nabla\cdot)\right)v,$$ and in particular, this seems very difficult to compute since the expressions for $F$ and $G$ are quite complicated.

My question is: does there exists a way of ensuring the existence for solution to my problem in an easy way, i.e, without looking for the kernel of $P^*_u$?

I have a particular guess that this can be done, how so?