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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 $$\lambda $$ λ -Wasserstein metric topology with $$1\le \lambda <\infty $$ 1 ≤ λ < ∞ , provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of $$\infty $$ ∞ -Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.

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