POSITIVE SOLUTION OF CRITICAL HARDY-SOBOLEV ELLIPTIC SYSTEMS WITH THE BOUNDARY SINGULARITY

In this paper, we are concerned with the existence of positive solutions to the system { -Delta u = 2p/p+q u(p-1) v(q) + 2 lambda alpha/alpha+beta u(alpha-1) v(beta)/vertical bar x vertical bar(s) , in Omega -Delta v = 2q/p+q u(p) v(q-1) + 2 lambda alpha/alpha+beta u(alpha) v(beta-1)/vertical bar x vertical bar(s) , in Omega (0.1) u > 0, v > 0, in Omega u = v = 0, on partial derivative Omega, where Omega is a C-2 domain in R-N with 0 epsilon partial derivative Omega, 0 < s < 2, lambda >0, p + q =2* = 2N/N-2, alpha + beta = 2*(s) = 2(N s)/N-2, N >= 3. We show that if Omega = R-+(N), problem (0.1) possesses a least energy solution and if Omega is bounded, 0 epsilon partial derivative Omega, there exists lambda* > 0 such that problem (0.1) has at least a positive solution provided 0 < lambda < lambda*.