  Nonlinear periodic systems, such as photonic crystals and Bose-Einstein
condensates (BECs) loaded into optical lattices, are often described by the
nonlinear Schr\"odinger/Gross-Pitaevskii equation with a sinusoidal potential.
Here, we consider a model based on such a periodic potential, with the
nonlinearity (attractive or repulsive) concentrated either at a single point or
at a symmetric set of two points, which are represented, respectively, by a
single {\delta}-function or a combination of two {\delta}-functions. This model
gives rise to ordinary solitons or gap solitons (GSs), which reside,
respectively, in the semi-infinite or finite gaps of the system's linear
spectrum, being pinned to the {\delta}-functions. Physical realizations of
these systems are possible in optics and BEC, using diverse variants of the
nonlinearity management. First, we demonstrate that the single
{\delta}-function multiplying the nonlinear term supports families of stable
regular solitons in the self-attractive case, while a family of solitons
supported by the attractive {\delta}-function in the absence of the periodic
potential is completely unstable. We also show that the {\delta}-function can
support stable GSs in the first finite gap in both the self-attractive and
repulsive models. The stability analysis for the GSs in the second finite gap
is reported too, for both signs of the nonlinearity. Alongside the numerical
analysis, analytical approximations are developed for the solitons in the
semi-infinite and first two finite gaps, with the single {\delta}-function
positioned at a minimum or maximum of the periodic potential. In the model with
the symmetric set of two {\delta}-functions, we study the effect of the
spontaneous symmetry breaking of the pinned solitons. Two configurations are
considered, with the {\delta}-functions set symmetrically with respect to the
minimum or maximum of the potential.
