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 ℝ n with partial singular weight: ( − Δ ) s u γ 1 | x ′ 2 = ∗ β + η α v , (0.1)where ∈ (0, 1), 0 ≤ α, < 2s n, m × η1, η2 > 1, : / s, γ1, γ2 γH, and H is some explicit constant. By establishing new embedding results partially weighted Morrey norms product space ˙ ), provide sufficient conditions under which a weak nontrivial solution of (0.1) exists via variational methods. We also extend these to p-Laplacian especially.