Some remarks on nonexistence in the Gamow liquid drop model

In this paper we deal with the celebrated Gamow liquid drop model, namely, for $m>0$, $N\ge 2$ and $\alpha \in (0,N)$, we study the problem $$\min \{P(\mathrm{\Omega })+V_{\alpha }(\mathrm{\Omega }):\mathrm{\Omega }\subset {\mathbb{R}}^N,\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }=m\},$$ and focus mostly on the nonexistence of minimizers for large masses. First we show that any possible minimizer in the asymptotic regime must satisfy an optimal thickness property. Eventually, in the simplified setting of sets made up as a tiling of cubes, we show that nonexistence occurs for large masses for the whole range of parameters $N\ge 2$ and $\alpha \in (0,N)$.