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We prove the existence of non-trivial global minimizers of a class of freeenergies related to aggregation equations with degenerate diffusion on$\Real^d$. Such equations arise in mathematical biology as models for organismgroup dynamics which account for competition between the tendency to aggregateinto groups and nonlinear diffusion to avoid over-crowding. The existence ofnon-zero optimal free energy stationary solutions representing coherent groupsin $\Real^d$ is therefore of interest. The primary contribution is theinvestigation of a notion of criticality associated with the global minimizerproblem. The notion arises from the scaling of diffusive and aggregative forcesas mass spreads and is shown to dictate the existence, and sometimesnon-existence, of global minimizers.

Global Minimizers for Free Energies of Subcritical Aggregation Equations with Degenerate Diffusion