The optical bound state in the continuum (BIC) is characterized by an extremely high-quality factor resulting in a drastic enhancement of light-matter interaction phenomena. We study the optical response of a one-dimensional photonic crystal slab with Kerr focusing nonlinearity in the vicinity of BICs analytically and numerically. We predict a strong nonlinear response including multistable behavior, self-tuning of BICs to the frequency of incident waves, and breaking of symmetry protected BICs. We show that all of these phenomena can be observed in silicon photonic structure at the pump power of several μW/cm2. We also analyze the modulation instability of the obtained solutions and the effect of the finite size of the structure on the stability. Our findings have strong implications for nonlinear photonics and integrated optical circuits.