  We study the nonlinear Schr\"odinger equation with a periodic delta-function
potential. This realizes a nonlinear Kr\"onig-Penney model, with physical
applications in the context of trapped Bose-Einstein condensate alkaly gases
and in the transmission of signals in optical fibers. We find analytical
solutions of zero-current Bloch states. Such wave-functions have the same
periodicity of the potential, and, in the linear limit, reduce to the Bloch
functions of the Kr\"onig-Penney model. We also find new classes of solutions
having a periodicity different from that of the external potential. We
calculate the chemical potential of such states and compare it with the linear
excitation spectrum.
