In Yau and Schoen's differential geometry,in Ch5 before Thm 3.5,the author says


When $R$(scalar curvature of a manifold M)$>0$,there exists a unique Green's function $G$ to the operator $L=-\Delta+aR$ and $LG\_{P}=\delta\_{P}$.Here $P$ is an arbitrary point of $M$.


In the normal coordinate of $P$,we have
$$
G\_P(x)=\frac{1}{(n-2) \omega\_{n-1}} r^{2-n}(1+o(1)).
$$


I wonder where can I find this existence result and how to get the expansion at point $P$.



Existence and estimates of Green's function on Riemannian manifold
For further details on the existence argument, see Chapter 4 of Aubin's book, "Nonlinear Analysis on Manifolds: Monge-Ampère Equations" [1] for the harmonic case, and [2] for the metaharmonic case. The asymptotic expansion formula near $P$ is derived from the first term of the Green function, which is obtained by multiplying the fundamental solution in $\mathbb{R}^n$ by a function that equals 1 near $P$.


[1] T. Aubin, "Nonlinear Analysis on Manifolds: Monge-Ampère Equations," Springer-Verlag New York, 1982.
[2] Yanir A. Rubinstein, "Lecture 16: Metaharmonic Functions," University of Maryland, 2019. Available at [http://www.math.umd.edu/~yanir/742/742-16-19.pdf](http://www.math.umd.edu/%7Eyanir/742/742-16-19.pdf).


