A spectral shape optimization problem with a nonlocal competing term

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