Multiple positive solutions of the stationary Keller-Segel system

Bonheure, Denis; Casteras, Jean-Baptiste; Noris, Benedetta

Document type :

Pré-publication ou Document de travail

Title :

Multiple positive solutions of the stationary Keller-Segel system

Author(s) :

Bonheure, Denis [Auteur]
Département de mathématiques Université Libre de Bruxelles
Quantitative methods for stochastic models in physics [MEPHYSTO]
Casteras, Jean-Baptiste [Auteur]
Quantitative methods for stochastic models in physics [MEPHYSTO]
Département de Mathématique [Bruxelles] [ULB]
Noris, Benedetta [Auteur]
Département de mathématiques Université Libre de Bruxelles

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

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33 pages

Collections :

Laboratoire Paul Painlevé - UMR 8524

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1603.07374
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