*Published Paper*

**Inserted:** 15 dec 2021

**Last Updated:** 18 apr 2023

**Journal:** Nonlinear Analysis

**Volume:** 231

**Pages:** Paper no. 113046

**Year:** 2023

**Doi:** https://doi.org/10.1016/j.na.2022.113046

**Abstract:**

This paper deals with ground states for systems governed by generalized Lennard-Jones potentials $LJ^{p,q}(r):= r^{-p} - r^{-q}$, for $0<q<1<p$. The energy per particle diverges to $-\infty$ as the number $N$ of particles diverges. As a consequence, the average distance between particles vanishes as $N\to +\infty$. After suitable scaling, we prove that such a model converges, as $N\to +\infty$ and in the sense of $\Gamma$-convergence, to a rotating stars model; the effective energy is given by the sum of a repulsive pressure term and an attractive nonlocal interaction functional. The ground states of such a limit energy have non constant density. As a consequence, for the generalized Lennard-Jones potentials considered here, crystallization does not occur in any reasonable sense.

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