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Steady state solutions of a reaction-diffusion system modeling chemotaxis. (English) Zbl 1002.35049
Summary: We study the following nonlinear elliptic equation \[ \begin{cases} \Delta u-\beta u+\lambda\left( {e^u\over\int_\Omega e^u}-{1\over |\Omega |}\right) =0\quad &\text{in }\Omega,\\ {\partial u\over\partial \nu}= 0\quad &\text{on }\partial \Omega\end{cases}, \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^2\). This equation arises in the study of stationary solutions of a chemotaxis system proposed by Keller and Segel. Under the condition that \(\beta> {\lambda\over |\Omega |}-\lambda_1\), \(\lambda\neq 4\pi m\) for \(m=1,2,\dots,\) 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.

35J65 Nonlinear boundary value problems for linear elliptic equations
35Q80 Applications of PDE in areas other than physics (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
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