Sobolev functions without compactly supported approximations

A basilar property and a useful tool in the theory of Sobolev spaces is density smooth compactly supported functions space $W^{k,p}(\R^n)$ (i.e. with weak derivatives orders $0$ to $k$ $L^p$). On Riemannian manifolds, it well known that same remains valid under suitable geometric assumptions. However, on complete non-compact manifold can fail be true general, as we prove this paper. This settles an open problem raised for instance by E. Hebey [\textit{Nonlinear analysis manifolds: inequalities}, Courant Lecture Notes Mathematics, vol. 5, 1999, pp. 48-49].