Fractional elliptic systems with critical nonlinearities

This paper deals with existence, uniqueness and multiplicity of positive solutions to the following nonlocal system of equations: {(-Delta)(s)u = alpha/2(s)*vertical bar u vertical bar(alpha-2)u vertical bar nu vertical bar(beta) + f(x) in R-N, (-Delta)(s)v = beta/2(s)*vertical bar nu vertical bar(beta-2)u vertical bar u vertical bar(alpha) + g(x) in R-N, (S) u, v > 0 in R-N, where 0 < s < 1, N > 2s, alpha, beta > 1, alpha + beta = 2N/(N - 2s), and f, g are nonnegative functionals in the dual space of (H)over dot(s)(R-N), i.e., ((H)over dot)s())' < f, u > (s)((H)over dots) >= 0, whenever u is a nonnegative function in (H)over dot(s)(R-N). When f = 0 = g, we show that the ground state solution of (S) is unique. On the other hand, when f and g are nontrivial nonnegative functionals with ker( f) = ker(g), then we establish the existence of at least two different positive solutions of (S) provided that parallel to f parallel to(s)((H)over dot)' and parallel to g parallel to(s)((H)over dot)' are small enough. Moreover, we also provide a global compactness result, which gives a complete description of the Palais-Smale sequences of the above system.