  The Patlak-Keller-Segel equation is a canonical model of chemotaxis to
describe self-organized aggregation of organisms interacting with chemical
signals. We investigate a variant of this model, assuming that the organisms
exert effective pressure proportional to the number density. From the resulting
set of partial differential equations, we derive a Lyapunov functional that can
also be regarded as the free energy of this model, and minimize it with a Monte
Carlo method to detect the condition for self-organized aggregation. Focusing
on radially symmetric solutions on a two-dimensional disc, we find that the
chemical interaction competes with diffusion so that aggregation occurs when
the relative interaction strength exceeds a certain threshold. Based on the
analysis of the free-energy landscape, we argue that the transition from a
homogeneous state to aggregation is abrupt yet continuous.
