A mass supercritical and critical Sobolev fractional Schrodinger system

We study the following coupled fractional Schrodinger system: {(-Delta)(s)u = lambda(1)u + mu(1)vertical bar u vertical bar(p-2)u + beta r(1)vertical bar u vertical bar(r1-2)u vertical bar v vertical bar(r2) in R-N, (-Delta)(s)v = lambda(2)v + mu(2)vertical bar v vertical bar(q-2)v + beta r(2)vertical bar u vertical bar(r1)vertical bar nu vertical bar(r2-2) v in R-N, with prescribed masses integral(RN) vertical bar u vertical bar(2)dx = a and integral(RN)vertical bar v vertical bar(2)dx = b. Here a, b > 0 are prescribed, N >= 2, 1/2 <= S < 1, and mu(1), mu(2), beta, are all positive constants. In this system, lambda(1), lambda(2) is an element of R are unknown and appear as Lagrange multipliers. Any (u, v) solving such system (for some lambda(1), lambda(2)) is called the normalized solution in the literature, where the normalization is settled in L-2 (R-N). For this model case of r(1) , r(2) > 1 and 2 + 4s/N < p, q, r(1) + r(2) < 2(s)* : = 2N / (N - 2s) with dimension 2 <= N <= 4s, for sufficiently large beta > 0, we show that there exists a positive normalized solution. We also show that, in the case of r(1) , r(2) > 1 and p = q = r(1) + r(2) = 2(s)*, the nonexistence of positive normalized solution.