Existence of symmetric solutions for singular semilinear elliptic systems with critical Hardy-Sobolev exponents

This paper deals with the singular semilinear elliptic system -Delta u = mu u/vertical bar x vertical bar(2) + 2 alpha Q(x)/(alpha + beta) vertical bar x vertical bar(s) vertical bar u vertical bar(alpha-2)u vertical bar v vertical bar beta + lambda vertical bar u vertical bar(q-2)u in Omega, -Delta v = mu v/vertical bar x vertical bar(2) + 2 beta Q(x)/(alpha + beta)vertical bar x vertical bar(s) vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2)v+delta vertical bar v vertical bar(q-2)v in Omega, u = v = 0 on partial derivative Omega, where Omega subset of R-N (N >= 3) is a smooth bounded domain, 0 epsilon Omega and Omega is G-symmetric with respect to a subgroup G of O(N), 0 <= mu < <(mu)over bar> with (mu) over bar = (N-2/2)(2), lambda, delta >= 0, 0 <= s < 2 and alpha, beta > 1 satisfy alpha + beta = 2*(s) = 2(N-s)/N-2, 2 < q < 2* = 2N/N-2, Q(x) is continuous and G-symmetric on (Omega) over bar. Based upon the symmetric criticality principle of Palais and variational methods, we obtain several existence results of G-symmetric solutions under certain appropriate hypotheses on Q, q and the pair of parameters (lambda, delta) epsilon R-2. (C) 2012 Elsevier Ltd. All rights reserved.