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

We consider the boundary value problem{−Δu+u=λeu,inΩ∂νu=0on∂Ω where Ω is a bounded smooth domain in R2R2, λ>0λ>0 and ν is the inner normal derivative at ∂Ω. This problem is equivalent to the stationary Keller–Segel system from chemotaxis.We establish the existence of a solution uλuλ which exhibits a sharp boundary layer along the entire boundary ∂Ω as λ→0λ→0. These solutions have large mass in the sense that ∫Ωλeuλ∼|log⁡λ|∫Ωλeuλ∼|log⁡λ|.