Large amplitude stationary solutions to a chemotaxis system. (English) Zbl 0676.35030

The existence of nonconstant stationary solutions to the Keller-Segel model [J. Theor. Biol. 26, 399-415 (1970)] is treated. The problem is reduced to the following single equation: \[ (1)\quad d\Delta w-w+w^ p=0\quad on\quad \Omega,\quad \partial w/\partial n=0\quad on\quad \partial \Omega, \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n.\)
It is shown that for d sufficiently small no nonconstant solutions to (1) are possible and for d sufficiently close to 0 there exist uniformly bounded nonconstant solutions. The behaviour of the latter is investigated when \(d\to 0\).


35J65 Nonlinear boundary value problems for linear elliptic equations
35B35 Stability in context of PDEs
92Cxx Physiological, cellular and medical topics
35B32 Bifurcations in context of PDEs
Full Text: DOI


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