A family of explicit minimizers for interaction energies

In this paper we consider the minimizers of the interaction energies with the power-law interaction potentials W ( x ) = | x | a a − | x | b b in d dimensions. For odd d with ( a , b ) = ( 3 , 2 − d ) and even d with ( a , b ) = ( 3 , 1 − d ) , we give the explicit formula for the unique energy minimizer up to translation. For the odd dimensions, the key observation is that successive Laplacian of the Euler–Lagrange condition gives a local partial differential equation for the minimizer. For the even dimensions d , the minimizer is given as the projection and rescaling of the previously constructed minimizer in dimension d + 1 via a new lemma on dimension reduction.