Fractional Choquard equation with critical nonlinearities

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving the fractional Laplacian (-Delta)(s) u = (integral(Omega) vertical bar u vertical bar(2)*(mu,s)/vertical bar x - y vertical bar(mu) dy) vertical bar u vertical bar(2)*(mu,s-2) u + lambda u in Omega, u = 0 in R-n\Omega, where Omega is a bounded domain in R-n with Lipschitz boundary, lambda is a real parameter, s is an element of (0, 1), n > 2s, 0 < mu < n and 2(mu,s)* = (2n - mu)/(n - 2s) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, nonexistence and regularity results for weak solution of the above problem using variational methods.