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