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

Abstract

This paper is devoted to the analysis of a degenerate Keller–Segel model with the diffusion exponent 2dd+2<m<2−2d and m>2−2d. For m≤2−2d, this model possesses a scaling invariant space Lp norm with p:=d(2−m)2. When m=2dd+2, a result of Chen et al. (2012) shows that the Lp norm of the steady states is the critical point of the free energy. For m=2−2d, the Lp norm of the steady states minimizes the free energy (Blanchet et al., 2009). In this paper, we will explore the relationship between the Lp norm of the steady states and the free energy with the diffusion exponent 2dd+2<m<2−2d. Here a modified Hardy–Littlewood–Sobolev inequality plays an important role: ∬Rd×Rdu(x)u(y)|x−y|d−2dxdy≤CHLS‖u‖p2−m‖u‖mm. In addition, we will give the upper bound of the support of the steady state solutions for the subcritical case m>2−2d via the concentration–compactness principle (Lions, 1984).