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.
There is also a constructive proof due to Li and Tam, which is easier to read for geometers.
