Fractional elliptic systems with critical nonlinearities

Elliptic systems arise in biological applications (e.g. population dynamics) or physical applications (e.g. models of a nuclear reactor) and have been drawn a lot of attention. In the nonlocal case, there are not so many papers on weakly coupled systems in R^N. In this paper we present some recent existence, uniqueness and multiplicity results for positive solutions of a class of weakly coupled nonlocal systems of equations in R^N, which are new also in the local case. Moreover, we also provide a global compactness result, which gives a complete description of the Palais-Smale sequences of the treated systems. To the best of our knowledge, this decomposition has been studied only for systems of equations in bounded domains.