Stationary solutions to a Keller–Segel chemotaxis system

Article type: Research Article

Authors: Musso, Monica | Wei, Juncheng

Affiliations: Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avda. Vicuña Mackenna 4860, Macul, Chile and Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy | Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong E-mail: wei@math.cuhk.edu.hk

Abstract: 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.

Journal: Asymptotic Analysis, vol. 49, no. 3-4, pp. 217-247, 2006