Let $(M,g)$ be a compact Riemannian manifold of dimension $n$ and $\theta$ a differential 1-form. Write $\theta=dg+\delta\mu+h$, the decomposition of $\theta$ according to the Hodge decomposition. I came across the following equation
$$\Delta(f-g)=\frac{n-2}{2(n-1)}g(df,df-\theta)$$where $f$ is the unknown and $\Delta$ is the Laplacian associated to the metric.

Is there some general result which ensures the existence of a solution of this equation?