Optimization problems with non-local repulsion

We review some optimization problems where an aggregating term is competing with a repulsive one, such as the Gamow liquid drop model, the Lord Rayleigh model for charged drops, and the ground state energy for the Hartree equation. As an original contribution, we show that for large values of the mass constraint, the ball is an unstable critical point of a functional made up as the sum of the first eigenvalue of the Dirichlet-Laplacian plus a Riesz-type repulsive energy term, in support to a recent open question raised in [MR21]