Droplet breakup in the liquid drop model with background potential

We consider a variant of Gamow’s liquid drop model, with a general repulsive Riesz kernel and a long-range attractive background potential with weight [Formula: see text]. The addition of the background potential acts as a regularization for the liquid drop model in that it restores the existence of minimizers for arbitrary mass. We consider the regime of small [Formula: see text] and characterize the structure of minimizers in the limit [Formula: see text] by means of a sharp asymptotic expansion of the energy. In the process of studying this limit we characterize all minimizing sequences for the Gamow model in terms of “generalized minimizers”.