Abstract

We consider the boundary value problem {-Δu+u-λeu=0,u>0inB1(0)∂νu=0on∂B1(0),whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here B1(0) is the unit disk, ν the outer normal to ∂B1(0) , and λ> 0 is a parameter. We show that, provided λ is sufficiently small, there exists a family of radial solutions uλ to this system which blow up at the origin and concentrate on ∂B1(0) , as λ→ 0. These solutions satisfy limλ→0uλ(0)|lnλ|=0and0