A nonlocal isoperimetric problem with density perimeter

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent $\alpha$, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter $\gamma$. We show that for a wide class of density functions the energy admits a minimizer for any value of $\gamma$. Moreover these minimizers are bounded. For monomial densities of the form $|x|^p$ we prove that when $\gamma$ is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the $\gamma\to 0$ limit corresponds, under a suitable rescaling, to a small mass $m=|\Omega|\to 0$ limit when $pd-\alpha+1$.