A note on the free energy of the Keller–Segel model for subcritical and supercritical cases

This paper is devoted to the analysis of a degenerate Keller–Segel model with the diffusion exponent 2dd+22−2d. For m≤2−2d, this model possesses a scaling invariant space LpLp norm with p:=d(2−m)2. When m=2dd+2, a result of Chen et al. (2012) shows that the LpLp norm of the steady states is the critical point of the free energy. For m=2−2d, the LpLp norm of the steady states minimizes the free energy (Blanchet et al., 2009). In this paper, we will explore the relationship between the LpLp norm of the steady states and the free energy with the diffusion exponent 2dd+22−2d via the concentration–compactness principle (Lions, 1984).