We numerically study light propagation through a specially designed nonlinear nanoscale metal-dielectric multilayer structure with a linear effective dielectric constant just below zero. The calculated dependence of the output intensity on the input intensity shows a steplike behavior. It rests upon an intensity-dependent change of the effective dielectric constant from negative (low-transmission state) to positive (high-transmission state) values, corresponding to a transition of the optical properties from metalliclike to dielectriclike. The study of the transient behavior of the structure demonstrates a switching time of around 1 ps.