Most of the current literature I've seen is either for compact Riemannian manifolds or unbounded subsets of Euclidean space. In this article, the authors consider a priori bounds on such systems on asymptotically Euclidean manifolds but don't consider existence. Is there a good reference for the existence theory?
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I've also noticed a seeming lack of literature on this topic, and even spent several years searching. What eventually proved helpful, and you may have seen, the convexity methods described in section 10.6 of Hormander, The Analysis of Linear Partial Differential Operators II, and as further elaborated, for instance, in section 4.7 of B. E. Petersen, Introduction to the Fourier Transform and Pseudo-differential Operators. These are stated for subsets of Euclidean space, but unless I'm mistaken, the functional-analytic arguments easily carry over to manifolds. To use the appropriate version of convex hull, one also needs a certain topological lemma, which I happened to come across at this post. I have a brief, unpublished summary of the result that you can find here.
