On a critical Schrodinger system involving Hardy terms

This paper concerns an elliptic system with critical exponents: {-Delta uj - lambda(j)/|x|(2)u(j) = u(j)(2)*(-1) + Sigma(k not equal j)beta(jk)alpha(jk)u(j)(alpha jk-1)u(k)(alpha kj), x is an element of R-N, u(j) is an element of D-1,D-2(R-N), u(j) > 0 in R-N \ {0}, j = 1, ... r where N >= 3, r >= 2,2* = 2N/N-2, lambda(j) is an element of(0, (N-2)(2)/4) for all j=1, ..., r; b(jk) = beta(kj); alpha(jk) > 1, alpha(kj) > 1, and alpha(jk) + alpha(kj) = 2* for all k not equal j. Note that the nonlinearities u(j)(2)*(- 1) and the coupling terms are all critical in arbitrary dimension N >= 3. The signs of the coupling constants beta(ij) are decisive for the existence of the ground-state solutions. We show that the critical system with r >= 3 has a positive ground-state solution for all beta(jk) > 0 with some constraint on lambda(j). However, there is no ground-state solution when all beta(jk) are negative. It is also proved that the positive solution of the system is radially symmetric. Furthermore, we obtain an uniqueness theorem for the case r >= 3 with N = 4 and an existence theorem for the case r = 2 with general coupling exponents.