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

We consider the boundary value problem $-\mathrm{\Delta }u+u=\lambda e^u$ in $\mathrm{\Omega }$ with Neumann boundary condition, where $\mathrm{\Omega }$ 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 $\mathrm{\partial }\mathrm{\Omega }$ as $\lambda \to 0$. These solutions have large mass in the sense that ${\int }_{\mathrm{\Omega }}\lambda e^{u_{\lambda }}\sim \mathrm{\mid }\log \lambda \mathrm{\mid }\mathrm{.}$