Periodic patterns for a model involving short-range and long-range interactions

We consider a physical model where the total energy is governed by surface tension and repulsive screened Coulomb potential on the 3-dimensional space. We obtain different periodic equilibrium patterns i.e. stationary sets for this energy, under some volume constraints. The patterns bifurcate smoothly from straight lamellae, lattices of round solid cylinders and lattices of round balls.