  We study numerically the effect of mode mixing and direct dipole-dipole
interactions between gain molecules on spasing in a small composite
nanoparticles with a metallic core and a dye-doped dielectric shell. By
combining Maxwell-Bloch equations with Green's function formalism, we calculate
lasing frequency and threshold population inversion for various gain densities
in the shell. We find that gain coupling to nonresonant plasmon modes has a
negligible effect on spasing threshold. In contrast, the direct dipole-dipole
coupling, by causing random shifts of gain molecules' excitation frequencies,
hinders reaching the spasing threshold in small systems. We identify a region
of parameter space in which spasing can occur considering these effects.
