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

Abstract

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