  We give some classifications of biharmonic hypersurfaces with constant scalar
curvature. These include biharmonic Einstein hypersurfaces in space forms,
compact biharmonic hypersurfaces with constant scalar curvature in a sphere,
and some complete biharmonic hypersurfaces of constant scalar curvature in
space forms and in a non-positively curved Einstein space. Our results provide
additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture
that a biharmonic submanifold in a sphere has constant mean curvature, and two
more cases that support Chen's conjecture on biharmonic hypersurfaces
(Corollaries 2.2,2.7).
