  We present a theory which explains how to achieve an enhancement of nonlinear
effects in a thin layer of nonlinear medium by involving a planar periodic
structure specially designed to bear a trapped-mode resonant regime. In
particular, the possibility of a nonlinear thin metamaterial to produce the
bistable response at a relatively low input intensity due to a large quality
factor of the trapped-mode resonance is shown. Also a simple design of an
all-dielectric low-loss silicon-based planar metamaterial which can provide an
extremely sharp resonant reflection and transmission is proposed. The designed
metamaterial is envisioned for aggregating with a pumped active medium to
achieve an enhancement of quantum dots luminescence and to produce an
all-dielectric analog of a 'lasing spaser'.
