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

Abstract

We consider the boundary value problem $$\begin{aligned}\left\{ \begin{array}{rcll} -\Delta u+ u -\lambda e^u&{}=&{}0,\ u>0 &{} \mathrm {in}\ B_1(0)\\ \partial _\nu u&{}=&{}0&{}\mathrm {on}\ \partial B_1(0), \end{array}\right. \end{aligned}$$whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here \(B_1(0)\) is the unit disk, \(\nu \) the outer normal to \(\partial B_1(0)\), and \(\lambda >0\) is a parameter. We show that, provided \(\lambda \) is sufficiently small, there exists a family of radial solutions \(u_\lambda \) to this system which blow up at the origin and concentrate on \(\partial B_1(0)\), as \(\lambda \rightarrow 0\). These solutions satisfy $$\begin{aligned}\lim _{\lambda \rightarrow 0} \frac{u_\lambda (0)}{|\ln \lambda |}=0\quad \text{ and }\quad 0<\lim _{\lambda \rightarrow 0} \frac{1}{|\ln \lambda |}\int _{B_1(0)}\lambda e^{u_\lambda (x)}dx<\infty , \end{aligned}$$having in particular unbounded mass, as \(\lambda \rightarrow 0\).