We consider the boundary value problem −Δu+u=λeu in
Ω with Neumann boundary condition, where Ω is a bounded smooth
domain in R2, λ>0. This problem is equivalent to the
stationary Keller-Segel system from chemotaxis. We establish the existence of a
solution uλ which exhibits a sharp boundary layer along the entire
boundary ∂Ω as λ→0. These solutions have large mass in
the sense that $ \int_\Omega \lambda e^{u_\lambda} \sim |\log\lambda|.