The Poisson equation in homogeneous Sobolev spaces

We consider Poisson’s equation in an n-dimensional exterior domain G (n≥ 2) with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certain Lq(G)-spaces there exists a solution in the homogeneous Sobolev space S2,q(G), containing functions being local in Lq(G) and having second-order derivatives in Lq(G). Concerning the uniqueness of this solution we prove that the corresponding nullspace has the dimension n+1, independent of q.