On a critical elliptic system with concave-convex nonlinearities

In this paper, we consider the following elliptic system involving critical Sobolev and Hardy-Sobolev exponents {-Delta u = 2 alpha/2* vertical bar u vertical bar(alpha-2)u vertical bar v vertical bar(beta) + vertical bar u vertical bar(2*(s)-2)u/vertical bar x vertical bar(s) + a(x)vertical bar u vertical bar(q-2)u in Omega, -Delta v = 2 beta/2* vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2) + vertical bar v vertical bar(2*(s)-2)v/vertical bar x vertical bar(s) + b(x)vertical bar v vertical bar(q-2)v in Omega, u = v = 0 where Omega is a bounded domains in R-N satisfying some geometric condition, N >= 3,0 < s < 2, 1 < q < 2, 2* (s) := 2(N-s)/(N-2) and alpha, beta > 1 such that alpha + beta = 2* := 2* (0). Our main result asserts that, if N > max(4, left perpendicular2sright perpendicular + 2), then our problem has two disjoint and infinite sets of solutions. The present work may be seen as a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in Ambrosetti et al. (J Funct Anal 122:519-543, 1994) for an elliptic systems.