  We prove the strong unique continuation property for many-body Schr\"odinger
operators with an external potential and an interaction potential both in
$L^p_{\rm loc}(\mathbb{R}^d)$, where $p > \max(2d/3,2)$, independently of the
number of particles. With the same assumptions, we obtain the Hohenberg-Kohn
theorem, which is one of the most fundamental results in Density Functional
Theory.
