Multiple positive solutions of the stationary Keller-Segel system

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.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-01398922
Contributor : Jean-Baptiste Casteras <>
Submitted on : Friday, November 18, 2016 - 8:30:21 AM
Last modification on : Tuesday, July 3, 2018 - 11:48:04 AM

Links full text

Identifiers

  • HAL Id : hal-01398922, version 1
  • ARXIV : 1603.07374

Collections

Citation

Denis Bonheure, Jean-Baptiste Casteras, Benedetta Noris. Multiple positive solutions of the stationary Keller-Segel system. 2016. ⟨hal-01398922⟩

Share

Metrics

Record views

195