Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive-attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the $\lambda -$Wasserstein metric topology with $1\le \lambda <\mathrm{\infty }$, provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of $\mathrm{\infty }$-Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.