  We consider the boundary value problem $-\Delta u + u =\lambda e^u$ in
$\Omega$ with Neumann boundary condition, where $\Omega$ is a bounded smooth
domain in $\mathbb R^2$, $\lambda>0.$ This problem is equivalent to the
stationary Keller-Segel system from chemotaxis. We establish the existence of a
solution $u_\lambda$ which exhibits a sharp boundary layer along the entire
boundary $\partial\Omega$ as $\lambda\to 0$. These solutions have large mass in
the sense that $ \int_\Omega \lambda e^{u_\lambda} \sim |\log\lambda|.$
