Question about the second order linear elliptic PDE on closed manifold

Recently I see a question linear second order PDE

in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE $$\Delta u +B^{i}(x)u_{i}+C(x)u=f$$ is equal to the poisson equation $$\Delta u = f$$ But it's well-known that the solvability of poisson equation on compact manifold is $$\int_{M} f = 0$$ If $$C<0$$, for the $$\Delta u +B^{i}(x)u_{i}+C(x)u=f$$ we can easily have an $$C^{0}$$ estimate, by Calderon-Zygmund inequality we can have a $$H^{2,2}$$ estimate, then iterate this process I think we can get the solution but we don't need the $$f$$ to be $$\int_{M} f = 0$$ There must be something wrong, could someone help me.