Unbounded mass radial solutions for the Keller–Segel equation in the disk

We consider the boundary value problem - Δ u + u - λ e u = 0 , u > 0 in B 1 ( 0 ) ∂ ν u = 0 on ∂ B 1 ( 0 ) , whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here B 1 ( 0 ) is the unit disk, ν the outer normal to ∂ B 1 ( 0 ) , and λ > 0 is a parameter. We show that, provided λ is sufficiently small, there exists a family of radial solutions u λ to this system which blow up at the origin and concentrate on ∂ B 1 ( 0 ) , as λ → 0 . These solutions satisfy lim λ → 0 u λ ( 0 ) | ln λ | = 0 and 0 < lim λ → 0 1 | ln λ | ∫ B 1 ( 0 ) λ e u λ ( x ) d x < ∞ , having in particular unbounded mass, as λ → 0 .