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 \(\gamma \rightarrow 0\) limit corresponds, under a suitable rescaling, to a small mass \(m=|\Omega |\rightarrow 0\) limit when \(p<d-\alpha +1\), but to a large mass \(m\rightarrow \infty \) for powers \(p>d-\alpha +1\).

我们考虑在体积约束下由密度周长和Riesz类型与指数\（\ alpha \）的非局部相互作用之和给出的能量函数的最小化，其中非局部相互作用的强度由参数\（ \ gamma \）。我们表明，对于广泛的密度函数类别，能量对于\（\ gamma \）的任何值都允许一个极小值。此外，这些最小化器是有界的。对于形式为\（| x | ^ p \）的单项式密度，我们证明了当\（\ gamma \）足够小时，唯一的极小值由固定体积的球给出。与恒定密度的情况相反，这里\（\ gamma \ rightarrow 0 \）极限在适当的重新缩放后对应于\（p <d- \ alpha +1 \）时较小的质量\（m = | \ Omega | \ rightarrow 0 \）极限，但对应于较大的质量\（m \ rightarrow \ infty \）的幂\（p> d- \ alpha +1 \）。