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 2d/d+2 < m < 2 - 2/d and m > 2 - 2/d. For m <= 2 - 2/d, this model possesses a scaling invariant space L-p norm with p := d(2-m)/2. When m = 2d/d+2, a result of Chen et al. (2012) shows that the L-p norm of the steady states is the critical point of the free energy. For m = 2 - 2/d, the L-p norm of the steady states minimizes the free energy (Blanchet et al., 2009). In this paper, we will explore the relationship between the L-p norm of the steady states and the free energy with the diffusion exponent 2d/d+2 < m < 2 - 2/d. Here a modified Hardy-Littlewood-Sobolev inequality plays an important role: integral integral(Rd xRd) u(x) u(y)/vertical bar x - y vertical bar(d-2)dxdy <= C-HLS parallel to u parallel to(2-m)(p)parallel to u parallel to(m)(m). In addition, we will give the upper bound of the support of the steady state solutions for the subcritical case m > 2 - 2/d via the concentration-compactness principle (Lions, 1984). (C) 2015 Elsevier Ltd. All rights reserved.