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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 \(d\to 0\).

MSC:

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
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