  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.
