We investigate the problem of Poincaré duality for differential forms on bounded subanalytic submanifolds of Rn (not necessarily compact). We show that, when p is sufficiently close to 1 then the cohomology of such a submanifold is isomorphic to its singular homology. In the case where p is large, we show that Lp cohomology is dual to intersection homology. As a consequence, we can deduce that the cohomology is Poincaré dual to Lq cohomology, if p and q are Hölder conjugate to each other and p is sufficiently large.