On doubly critical coupled systems involving fractional Laplacian with partial singular weight

In this paper, we consider the doubly critical coupled systems involving fractional Laplacian in Double-struck capital Rn with partial singular weight: {(-Delta)(s)u - gamma(1)u/vertical bar x 'vertical bar 2s = vertical bar u vertical bar(2s*(beta)-2)u/vertical bar x 'vertical bar beta + eta(1)/2(s)*(alpha) vertical bar u vertical bar(eta 1-2)u vertical bar v vertical bar(eta 2)/vertical bar x 'vertical bar(alpha), (-Delta)(s)v - gamma(2)v/vertical bar x 'vertical bar(2s) = vertical bar v vertical bar(2s*(beta))-2(v)/vertical bar x 'vertical bar(beta) + eta(2)/2(s)*(alpha) vertical bar v vertical bar(eta 2-2)v vertical bar u vertical bar(eta 1)/vertical bar x 'vertical bar(alpha), (0.1) where s is an element of (0, 1), 0 <= alpha, beta < 2s < n, 0 < m < n, x=(x ',x '')is an element of R-m x Rn-m, eta(1), eta(2) > 1, eta(1) + eta(2) =2(s)*(alpha) := 2(n-alpha)/n - 2s, gamma(1), gamma(2) < gamma(H), and gamma(H) = gamma(H)(n, m, s) > 0 is some explicit constant. By establishing new embedding results involving partially weighted Morrey norms in the product space (H) overd dot(s)(R-n) x (H) overd dot(s)(R-n), we provide sufficient conditions under which a weak nontrivial solution of (0.1) exists via variational methods. We also extend these results to p-Laplacian systems especially.