A perturbed problem of elliptic system with critical exponent

This paper deals with the following perturbed problem with critical exponent {-Delta u = u(2*-1) + alpha/2*u(alpha-1)v(beta) + epsilon K(x)u(p), x is an element of R-N, -Delta v = v(2*-1) + beta/2*u(alpha)v(beta-1) + epsilon Q(x)v(q), x is an element of R-N, (1) u > 0, v > 0, x is an element of R-N, where 1 <= p, q < 2* - 1, alpha + beta = 2* := 2N/N-2, alpha, beta >= 2, N = 3, 4 and epsilon is a parameter. Using a perturbation argument and Lyapunov-Schmidt reduction method, we obtain the existence of positive solutions to problem (1) and the asymptotic property of the solutions.