  The existence of compactly supported global minimisers for continuum models
of particles interacting through a potential is shown under almost optimal
hypotheses. The main assumption on the potential is that it is catastrophic, or
not H-stable, which is the complementary assumption to that in classical
results on thermodynamic limits in statistical mechanics. The proof is based on
a uniform control on the local mass around each point of the support of a
global minimiser, together with an estimate on the size of the "gaps" it may
have. The class of potentials for which we prove existence of global minimisers
includes power-law potentials and, for some range of parameters, Morse
potentials, widely used in applications. We also show that the support of local
minimisers is compact under suitable assumptions.
