Belmonte Beitia, Juan. 

We address the Gross-Pitaevskii equation with a periodic linear potential and a periodic sign-varying nonlinearity coefficient. Contrary to the claims in the previous works, we show that the intersite cubic nonlinear terms in the discrete nonlinear Schrdinger (DNLS) equation appear beyond the applicability of assumptions of the tight-binding approximation. Instead of these terms, for an even linear potential and an odd nonlinearity coefficient, the DNLS equation and other reduced equations have the quintic nonlinear term, which correctly describes bifurcation of gap solitons in the semi-infinite gap.