  The ground-state phase diagram of $S=2$ antiferromagnetic Heisenberg chains
with coexisting uniform and alternating single-site anisotropies is
investigated by the numerical exact diagonalization and density matrix
renormalization group methods. We find the Haldane, large-$D$, N\'eel,
period-doubled N\'eel, gapless spin fluid, quantized and partial ferrimagnetic
phases. The Haldane phase is limited to the close neighborhood of the isotropic
point. Within numerical accuracy, the transition from the gapless spin-fluid
phase to the period-doubled N\'eel phase is a direct transition. Nevertheless,
the presence of a narrow spin-gap phase between these two phases is suggested
on the basis of the low-energy effective theory. The ferrimagnetic ground state
is present in a wide parameter range. This suggests the realization of
magnetized single-chain magnets with a uniform spin magnitude by controlling
the environment of each magnetic ion without introducing ferromagnetic
interactions.
