Multiple positive solutions of elliptic systems in exterior domains

In this paper, existence and multiplicity of positive solutions of the elliptic system {-Delta u + V-1(x)u = mu(1)(x)u(3) + beta(x)uv(2) in Omega(epsilon), -Delta v + V-2(x)v = beta(x)u(2)v + beta mu(2)(x)v(3) in Omega(epsilon,) u = v = 0 on partial derivative Omega(epsilon) is proved, where Omega(epsilon) is an exterior domain in R-N such that R-N\Omega(epsilon) is far away from the origin and contains a sufficiently large ball, N = 1,2,3, and the coefficients V-j, mu(j), beta are continuous functions on R-N which tend to positive constants at infinity. We do not assume mu(1), mu(2), beta to be positive functions.