  We study the minimization of a spectral functional made as the sum of the
first eigenvalue of the Dirichlet Laplacian and the relative strength of a
Riesz-type interaction functional. We show that when the Riesz repulsion
strength is below a critical value, existence of minimizers occurs. Then we
prove, by means of an expansion analysis, that the ball is a rigid minimizer
when the Riesz repulsion is small enough. Eventually we show that for certain
regimes of the Riesz repulsion, regular minimizers do not exist.
