  Bose-Einstein condensates (BECs) in periodic potentials generate interesting
physics on the instabilities of Bloch states. The lowest-energy Bloch states of
BECs in pure nonlinear lattices are dynamically and Landau unstable, which
breaks down BEC superfluidity. In this paper we propose to use an out-of-phase
linear lattice to stabilize them. The stabilization mechanism is revealed by
the averaged interaction. We further incorporate a constant interaction into
BECs with mixed nonlinear and linear lattices, and reveal its effect on the
instabilities of Bloch states in the lowest band.
