Multiple positive solutions of the stationary Keller–Segel system

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.