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We study global minimizers of a functional modeling the free energy of thinliquid layers over a solid substrate under the combined effect of surface,gravitational, and intermolecular potentials. When the latter ones have a mildrepulsive singularity at short ranges, global minimizers are compactlysupported and display a microscopic contact angle of $\pi/2$. Depending on theform of the potential, the macroscopic shape can either be droplet-like orpancake-like, with a transition profile between the two at zero spreadingcoefficient. These results generalize, complete, and give mathematical rigor tode Gennes' formal discussion of spreading equilibria. Uniqueness andnon-uniqueness phenomena are also discussed.

Spreading equilibria under mildly singular potentials: pancakes versus droplets