Steady state solutions of a reaction-diffusion system modeling chemotaxis. (English) Zbl 1002.35049

Wang, Guofang; Wei, Jun-Cheng

Math. Nachr. 233-234, 221-236 (2002).

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.

MSC:

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

Keywords:

Palais-Smale conditions; existence; Struwe’s technique; blow up analysis