Unbounded mass radial solutions for the Keller-Segel equation in the disk

Abstract

We consider the boundary value problem

{-Delta u + u - lambda e(u) = 0 ,u > 0 in B-1(0)

partial derivative(nu)u = 0 on partial derivative B-1(0),

whose solutions correspond to steady states of the Keller-Segel system for chemotaxis. Here B-1(0) is the unit disk,. the outer normal to partial derivative B-1(0), and lambda > 0 is a parameter. We show that, provided lambda is sufficiently small, there exists a family of radial solutions u(lambda) to this system which blow up at the origin and concentrate on partial derivative B-1(0), as lambda -> 0. These solutions satisfy

lim(lambda -> 0) u lambda(0)/vertical bar in lambda vertical bar = 0 and 0 lim(lambda -> 0) 1/vertical bar in lambda vertical bar integral(B1(0)) (lambda eu lambda(x)) dx < infinity,

having in particular unbounded mass, as lambda -> 0.