Solvability of a partial differential equation

Question

Suppose $(M,g)$ is a compact Riemannian manifold without boundary. Consider the following PDE on $M$: $$-\Delta_g u+|\nabla_g u|^2=f,$$ where $f$ is a given smooth function in $M$, $\Delta_g$ and $\nabla_g$ are the Laplacian and gradient of $g$ respectively. I wonder when it is solvable. Using integration by parts, I can derive the following necessary condition: $$\int_Mf=\int_M(-\Delta_g u+|\nabla_g u|^2)=\int_M|\nabla_g u|^2\geq 0,$$ that is, the integral of $f$ must be nonnegative. I wonder if it is also sufficient.