MiS Preprint Repository

We study the following nonlinear elliptic equation
$\{\delta u-\beta u+\lambda (\frac{e^u}{{\int }_{\omega }{e}^u}-\frac{1}{|\mathrm{\Omega }|})=0in\omega ,\frac{\delta u}{\delta v}=0on\delta \mathrm{\Omega }$
where $\mathrm{\Omega }$ 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 $\beta >\frac{\lambda }{|\mathrm{\Omega }|}-{\lambda }_1,\lambda \ne 4\pi m$ for m=1,2,..., where ${\lambda }_1$ is the first (nonzero) eigenvalue of $-\mathrm{\Delta }$ 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.