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

We consider the boundary value problem $-Δu + u =λe^u$ in $Ω$ with Neumann boundary condition, where $Ω$ is a bounded smooth domain in $\mathbb R^2$, $λ>0.$ This problem is equivalent to the stationary Keller-Segel system from chemotaxis. We establish the existence of a solution $u_λ$ which exhibits a sharp boundary layer along the entire boundary $\partialΩ$ as $λ\to 0$. These solutions have large mass in the sense that $\int_Ωλe^{u_λ} \sim |\logλ|.$

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