DIFFUSION VS CROSS-DIFFUSION : AN ELLIPTIC APPROACH

where 2= i=1 2 xi is the Laplace operator, 0 is a bounded smooth domain in R, 0 is the boundary of 0, and & is the outward unit normal vector on 0. The system (1.1) was proposed by Shigesada et al. [13] to model segregation of interacting species, where u and v represent the densities of two competing species, hence only non-negative u and v are of interest. The constants dj , aj , bj , and cj ( j=1, 2) are all positive, where d1 , d2 are the diffusion rates of these two species, a1 , a2 denote their intrinsic growth rates, b1 and c2 account for intra-specific competitions, and b2 and c1 are the coefficients of inter-specific competitions. The constants \12 and \21 are non-negative, and they are referred as cross-diffusion pressures. We refer to [13] for more background about (1.1). Previous work on the system (1.1) includes [2, 4 8, 15] and the references therein, and we refer to the introduction part of [4] for a more detailed description. In paper [4], we established various multidimensional existence and non-existence results on non-constant positive solutions of (1.1). Article ID jdeq.1998.3559, available online at http: www.idealibrary.com on