Multiple positive solutions of the stationary Keller-Segel system

TL;DR

This paper constructs multiple-layer radial solutions to the stationary Keller-Segel equation in a ball, revealing that these layers do not accumulate at the boundary but instead satisfy an optimal partition problem as the parameter approaches zero.

Contribution

It introduces a novel gluing variational method to build multi-layer solutions that concentrate on spheres, differing from previous results by showing layers do not accumulate at the boundary.

Findings

Multiple-layer solutions are constructed for the Keller-Segel system.

Solutions' layers satisfy an optimal partition problem in the limit.

Layers do not accumulate at the boundary as the parameter tends to zero.

Abstract

We consider the stationary Keller-Segel equation \begin{equation*} \begin{cases} -\Delta v+v=\lambda e^v, \quad v>0 \quad & \text{in }\Omega,\\ \partial_\nu v=0 &\text{on } \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a ball. In the regime $\lambda\to 0$, we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given natural positive number $n$, we build a solution having multiple layers at $r_1,\ldots,r_n$ by which we mean that the solutions concentrate on the spheres of radii $r_i$ as $\lambda\to 0$ (for all $i=1,\ldots,n$). A remarkable fact is that, in opposition to previous known results, the layers of the solutions do not accumulate to the boundary of $\Omega$ as $\lambda\to 0$. Instead they satisfy an optimal partition problem in the limit.