Mountain Pass Critical Points of the Liquid Drop Model

Abstract: We consider Gamow's liquid drop functional, $\mathcal{E}$, on $\mathbb{R}^3$ and construct non-minimizing, volume constrained, critical points for volumes $3.512 \cong \alpha_0 < V < 10$. In this range, we establish a mountain pass set up between a ball of volume $V$ and two balls of volume $V/2$ infinitely far apart. Intuitively, our critical point corresponds to the maximal energy configuration of an atom of volume $V$ as it undergoes fission into two atoms of volume $V/2$. Our proof relies on geometric measure theoretical methods from the min-max construction of minimal surfaces, and along the way, we address issues of non-compactness, ``pull tight" with a volume constraint, and multiplicity.