  This paper is concerned with volume-constrained minimization problems derived
from Gamow's liquid drop model for the atomic nucleus, involving the
competition of a perimeter term and repulsive nonlocal potentials. We consider
a large class of potentials, given by general radial nonnegative kernels which
are integrable on $\mathbb{R}^n$, such as Bessel potentials, and study the
behavior of the problem for large masses (i.e., volumes). Contrarily to the
small mass case, where the nonlocal term becomes negligible compared to the
perimeter, here the nonlocal term explodes compared to it. However, using the
integrability of those kernels, we rewrite the problem as the minimization of
the difference between the classical perimeter and a nonlocal perimeter, which
converges to a multiple of the classical perimeter as the mass goes to
infinity. Renormalizing to a fixed volume, we show that, if the first moment of
the kernels is smaller than an explicit threshold, the problem admits
minimizers of arbitrarily large mass, which contrasts with the usual case of
Riesz potentials. In addition, we prove that, any sequence of minimizers
converges to the ball as the mass goes to infinity. Finally, we study the
stability of the ball, and show that our threshold on the first moment of the
kernels is sharp in the sense that large balls go from stable to unstable. A
direct consequence of the instability of large balls above this threshold is
that there exist nontrivial compactly supported kernels for which the problems
admit minimizers which are not balls, that is, symmetry breaking occurs.
