Steady State Solutions of a Reaction-diffusion System Modeling Chemotaxis

We study the following nonlinear elliptic equation 8 < : u ? u + (e u R e u ? 1 jj) = 0 in ; @u @ = 0 on @; where 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 > jj ? 1 ; 6 = 4m for m = 1; 2; :::, where 1 is the rst (nonzero) eigenvalue of ? 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. 1. Introduction Chemotaxis is one of the simplest mechanisms for aggregation of biological species. The term refers to a situation where organisms (e.g., bacteria) move towards high concentrations of a chemical which they secrete. A basic model in chemotaxis was introduced by Keller and Segel 21]. They considered an advection-diiusion system consisting of two coupled parabolic equations for the concentration of species (which we shall denote by u(x; t)), and that of the chemical released (to be