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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.
