# Poisson equation on noncompact manifold

Let $$(M,g)$$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $$C^\infty_c$$-functions are dense in $$L^p_k$$ space.

For the equation $$\Delta u=f,$$ for some $$f\in L^2(M)$$.

Q How can we find a solution $$u\in L^2$$ satisfies the above equation?

Is there any related work?

In fact, for any noncompact Riemannian manifold $$M$$ (not necessarily with bounded geometry) and for any distribution $$f$$, there is a distribution $$u$$ such that $$Du=f$$, where $$D$$ is an elliptic operator satisfying the unique continuation property.
This was proved by Malgrange in Existence et approximation des solutions des ́equations aux d ́eriv ́ees partielles et des ́equations de convolution'. The Euclidean case was discussed in Hormander's bookLinear partial differential operators' (Chapter 4 as I remembered), and the arguments therein carries over to general Riemannian manifolds.