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). Contrary 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.

本文涉及从Gamow原子核液滴模型得出的体积受限的最小化问题，涉及周长项和排斥性非局部势的竞争。我们考虑了由一般可径向积分的径向非负核给出的一大类势能${\mathbb{[R}}^{ñ}$（例如Bessel势），并研究大质量（即体积）问题的行为。与小质量情况相反，在这种情况下，非局部项与周长相比可以忽略不计，此处非局部项与之相比会爆炸。但是，利用这些内核的可积性，我们将问题重写为经典周长和非局部周长之间的差异最小化，随着质量达到无穷大，收敛到经典周长的倍数。重新归一化为固定体积，我们表明，如果内核的第一矩小于显式阈值，则该问题允许任意大质量的最小化器，这与通常的Riesz势形成对比。此外，我们证明，随着质量达到无穷大，任何最小化序列都将收敛到球上。最后，我们研究了球的稳定性，并表明从大球从稳定到不稳定的意义上，我们对籽粒一刻的阈值是敏锐的。大于此阈值的大球的不稳定性的直接结果是，存在非平密的紧密支撑的内核，对于这些内核，问题允许使用非球的最小化器，即发生对称破坏。