N2 Law for bosons in 2D

A rigorous upper bound is derived for the exact ground-state energy of N negative charged bosons and N motionless, i.e. fixed, positive charges with Coulomb interactions in 2D for arbitrary N ⪖ 4 giving rise to an N2-upper bound. The consistency of such an N2 behaviour is also investigated by examining a lower bound to the ground-state energy.