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

Abstract

We consider the boundary value problem $$\{\begin{array}{rcll}-\Delta u+u-\lambda e^u& =& 0,u>0& \mathrm{in}{B}_1(0)\\ {\partial }_{\nu }u& =& 0& \mathrm{on}\partial B_1(0),\end{array}$$ 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 $\lambda \to 0$. These solutions satisfy $$ \lim_{λ\to 0} \frac{u_λ(0)}{|\lnλ|}=0\quad \mbox{and}\quad 0<\lim_{λ\to 0} \frac{1}{|\lnλ|}\int_{B_1(0)}λe^{u_λ(x)}dx<\infty, $$ having in particular unbounded mass, as $\lambda \to 0$.

33 pages. This is a major revision of the previous version, which contained a significant error. The final version will appear in Calculus of Variations and Partial Differential Equations

References in corpus (1)

• Large Conformal metrics with prescribed sign-changing Gauss curvature