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Multiple positive solutions of the stationary Keller–Segel system

  • Denis Bonheure
  • Jean-Baptiste Casteras
  • Benedetta NorisEmail author
Article

Abstract

We consider the stationary Keller–Segel equation
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta v+v=\lambda e^v, \quad v>0 \quad &{} \text {in }\Omega ,\\ \partial _\nu v=0 &{}\text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a ball. In the regime \(\lambda \rightarrow 0\), we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given \(n\in \mathbb {N}_0\), 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 \(\lambda \rightarrow 0\) (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 \(\lambda \rightarrow 0\). Instead they satisfy an optimal partition problem in the limit.

Mathematics Subject Classification

35J25 35B05 35B09 35B25 35B32 35B40 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Denis Bonheure
    • 1
  • Jean-Baptiste Casteras
    • 1
  • Benedetta Noris
    • 1
    Email author
  1. 1.Département de MathématiqueUniversité libre de BruxellesBrusselsBelgium

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