Large mass boundary condensation patterns in the stationary Keller-Segel system

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 $\int_\Omega \lambda e^{u_\lambda} \sim |\log\lambda|.$