Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (compactly-supported) smooth sections. Now, lets say I have given an *elliptic* linear differential operator $D:\Gamma^{\infty}(E)\to\Gamma^{\infty}(E)$. I am interested in the problem of finding solutions $u\in L^{2}(M)$ to the problem
$$Du=f$$
for a *compactly-supported source* $f\in \Gamma^{\infty}_{c}(M)$. Of course, if $M$ is compact everything is clear (cf. Fredholm theory), so I am asking specifically for the non-compact case (Lets say $M$ is complete, to be concrete). 

Of course, the question is very much related to the question whether a Green's function exists for $D$. I am aware of the paper of Malgrange (see 1. below), which states that any elliptic operator with constant-coefficient on $\mathbb{R}^{n}$ admits a Green's function, however, the original paper is in French and hence not accessible to me. A textbook version of this statement should also be somewhere contained in Hörmander, but only in the Euclidean case. There is this paper by Li-Tam (see 2. below), which give a more constructive proof (from a more differential geometric point of view) of Malgrange's theorem in the specific case of the Laplace-Beltrami operator. They do, however, state in the introduction of their paper that 

> In 1955 Malgrange studied elliptic operators on complete
> Riemannian manifolds which satisfied unique continuation property. In particular, he proved that the Laplace operator admits a symmetric Green's function [...].

But again, the original paper of Malgrange is not accessible to me and it is, as far as I know, it is for $\mathbb{R}^{n}$. So, let me summarise:

> Does anyone know a reference regarding the existence of $L^{2}$-solutions of elliptic problems on non-compact manifold with sources in $C^{\infty}_{c}$? In particular, what are the precise assumptions on the manifold (completeness I guess, any curvature bounds?) and operator (unique continuation property?)

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1. Malgrange: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. *Ann. Inst. Fourier (Grenoble)*, 6:271-355, 1955/6. (see [here][2] for an online version at Numdam)
2. Li, Tam: Symmetric Green's Functions on Complete Manifolds, American Jounral of Mathematics, 109(6):1129-1154, 1987. (See [here][1] for an online version at jstor)


  [1]: https://www.jstor.org/stable/2374588
  [2]: http://www.numdam.org/item/AIF_1956__6__271_0/