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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
52/1998

Steady state solutions of a reaction-diffusion system modeling chemotaxis

Guofang Wang and Jun-Cheng Wei

Abstract

We study the following nonlinear elliptic equation
$\{ \delta u- \beta u + \lambda (\frac{e^u}{\int_\omega e^u}-\frac{1}{|\Omega|})=0\ in \ \omega , \frac{\delta u}{\delta v} =0 \ on \ \delta \Omega$
where $\Omega$ is a smooth bounded domain in $R^2$. This equation arises in the study of stationary solutions of a chemotaxis system proposed by Keller and Segel. Under the condition that $\beta > \frac{\lambda}{|\Omega|} - \lambda_1 , \lambda \neq 4\pi m$ for m=1,2,..., where $\lambda_1$ is the first (nonzero) eigenvalue of $-\Delta$ under the Neumann boundary condition, we establish the existence of a solution to the above equation. Our idea is a combination of Struwe\'s technique and blow up analysis for a problem with Neumann boundary condition.

Received:
16.11.98
Published:
16.11.98

Related publications

inJournal
2002 Repository Open Access
Guofang Wang and Jun-Cheng Wei

Steady state solutions of a reaction-diffusion system modeling chemotaxis

In: Mathematische Nachrichten, 233/234 (2002), pp. 221-236