A mathematical model describing how a cell membrane with diffusive mobile receptors wraps around a ligand coated cylindrical or spherical particle has been recently developed to model particle size effects in receptor mediated endocytosis. The model predicted an optimal particle size for the smallest wrapping time, as well as a minimum and a maximum particle size for successful wrapping. The results showed that particles in the size range of tens to hundreds of nanometers can enter cells even in the absence of clathrin or caveolin coats. Here we report some further progresses on modeling a spontaneous membrane curvature induced by clathrin or caveolin coats at the inner membrane leaflet as well as simultaneous entry of many particles into cells. It is found that a spontaneous membrane curvature narrows the particle size range for successful wrapping and there exists an optimal particle density for maximum particle adsorptivity.