  We consider a model of Bose-Einstein condensates which combines a stationary
optical lattice (OL) and periodic change of the sign of the scattering length
(SL) due to the Feshbach-resonance management. Ordinary solitons and ones of
the gap type being possible, respectively, in the model with constant negative
and positive SL, an issue of interest is to find solitons alternating, in the
case of the low-frequency modulation, between shapes of both types, across the
zero-SL point. We find such \textit{alternate solitons} and identify their
stability regions in the 2D and 1D models. Three types of the dynamical regimes
are distinguished: stable, unstable, and semi-stable. In the latter case, the
soliton sheds off a conspicuous part of its initial norm before relaxing to a
stable regime. In the 2D case, the threshold (minimum number of atoms)
necessary for the existence of the alternate solitons is essentially higher
than its counterparts for the ordinary and gap solitons in the static model. In
the 1D model, the alternate solitons are also found only above a certain
threshold, while the static 1D models have no threshold. In the 1D case, stable
antisymmetric alternate solitons are found too. Additionally, we consider a
possibility to apply the variational approximation (VA) to the description of
stationary gap solitons, in the case of constant positive SL. It predicts the
solitons in the first finite bandgap very accurately, and does it reasonably
well in the second gap too. In higher bands, the VA predicts a border between
tightly and loosely bound solitons.
