Stationary solutions to a Keller–Segel chemotaxis system

We consider the following stationary Keller–Segel system from chemotaxis $\[\Delta u-au+u^{p}=0,\quad u>0\ \hbox{in}\ {\varOmega},\qquad \frac{\curpartial u}{\curpartial \nu}=0\quad \hbox{on}\ \curpartial {\varOmega},\]$ where Ω⊂ $\mathbb{R}$ ² is a smooth and bounded domain. We show that given any two positive integers K,L, for p sufficiently large, there exists a solution concentrating in K interior points and L boundary points. The location of the blow-up points is related to the Green function. The solutions are obtained as critical points of some finite-dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.