  Motivated by Gamow's liquid drop model in the large mass regime, we consider
an isoperimetric problem in which the standard perimeter $P(E)$ is replaced by
$P(E)-\gamma P_\varepsilon(E)$, with $0<\gamma<1$ and $P_\varepsilon$ a
nonlocal energy such that $P_\varepsilon(E)\to P(E)$ as $\varepsilon$ vanishes.
We prove that unit area minimizers are disks for $\varepsilon>0$ small enough.
More precisely, we first show that in dimension $2$, minimizers are necessarily
convex, provided that $\varepsilon$ is small enough. In turn, this implies that
minimizers have nearly circular boundaries, that is, their boundary is a small
Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we
prove that (in arbitrary dimension $n\geq 2$) the unit ball in $\mathbb{R}^n$
is the unique unit-volume minimizer of the problem among centered nearly
spherical sets. As a consequence, up to translations, the unit disk is the
unique minimizer. This isoperimetric problem is equivalent to a generalization
of the liquid drop model for the atomic nucleus introduced by Gamow, where the
nonlocal repulsive potential is given by a radial, sufficiently integrable
kernel. In that formulation, our main result states that if the first moment of
the kernel is smaller than an explicit threshold, there exists a critical mass
$m_0$ such that for any $m>m_0$, the disk is the unique minimizer of area $m$
up to translations. This is in sharp contrast with the usual case of Riesz
kernels, where the problem does not admit minimizers above a critical mass.
