Normalized solutions to the Schrodinger systems with double critical growth and weakly attractive potentials

In this paper, we look for solutions to the following critical Schrodinger system {-Delta u + (V-1 + lambda(1))u = vertical bar u vertical bar(2*-2)u + vertical bar u vertical bar(p1-2)u + beta r(1)vertical bar u vertical bar(r1-2)u vertical bar v vertical bar(r2) in R-N, -Delta v + (V-2 + lambda(2))v = vertical bar v vertical bar(2*-2)v + vertical bar v vertical bar(p2-2)v + beta r(2) vertical bar u vertical bar(r1) vertical bar v vertical bar(r2-2)v in R-N, having prescribed mass integral(RN) u(2) = a(1) > 0 and integral(RN) v(2) = a(2) > 0, where lambda(1), lambda(2) is an element of R will arise as Lagrange multipliers, N >= 3, 2* = 2N/(N-2) is the Sobolev critical exponent, r(1), r(2) > 1, p(1), p(2), r(1) + r(2) is an element of (2 + 4/N, 2*) and beta > 0 is a coupling constant. Under suitable conditions on the potentials V-1 and V-2, beta* > 0 exists such that the above Schrodinger system admits a positive radial normalized solution when beta >= beta*. The proof is based on comparison argument and minmax method.