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Multiple positive solutions of the stationary Keller–Segel system

par Bonheure, Denis ;Casteras, Jean-Baptiste ;Noris, Benedetta
Référence Calculus of variations and partial differential equations, 56, 3, 74, 35 pp.
Publication Publié, 2017-06

Article révisé par les pairs

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Résumé :

We consider the stationary Keller–Segel equation {-Δv+v=λev,v>0inΩ,∂νv=0on∂Ω,where Ω is a ball. In the regime λ→ 0 , we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given n∈ N0, we build a solution having multiple layers at r1, … , rn by which we mean that the solutions concentrate on the spheres of radii ri as λ→ 0 (for all i= 1 , … , n). A remarkable fact is that, in opposition to previous known results, the layers of the solutions do not accumulate to the boundary of Ω as λ→ 0. Instead they satisfy an optimal partition problem in the limit.