Multiple positive solutions of the stationary Keller-Segel system

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.