Collapsing steady states of the Keller-Segel system

Abstract

We consider the boundary value problem: Delta u - au + epsilon(2)e(u) = 0, u > 0 in Omega, sigma u/sigma v = 0 on sigma Omega,

which is equivalent to the stationary Keller-Segel system from chemotaxis. Here Omega subset of R-2 is a smooth and bounded domain. We show that given any two non-negative integers k, l with k + l >= 1, for e sufficiently small, there exists a solution u(epsilon) for which epsilon(2)e(u epsilon) develops asymptotically k interior Dirac deltas with weight 8 pi and l boundary deltas with weight 4 pi. Location of blow-up points is characterized explicitly in terms of Green's function of the Neumann problem.