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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.

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