Existence and Liouville theorems for coupled fractional elliptic system with Stein-Weiss type convolution parts

In this paper, we are mainly concerned with the following fractional coupled elliptic system with Stein-Weiss type convolution parts: {(-Delta)(S/2) u = c1(1/vertical bar x vertical bar(sigma) * u(p)/vertical bar x vertical bar(a)) u(q)/vertical bar x vertical bar(a) + beta(1/vertical bar x vertical bar(sigma) * v(p)/vertical bar x vertical bar(a))u(q)/vertical bar x vertical bar(a), in R-N, (-Delta)(S/2) v =c(2)(1 vertical bar x vertical bar(sigma) * v(p)/vertical bar x vertical bar(a)) + beta(1/vertical bar x vertical bar(sigma) * u(p)/vertical bar x vertical bar(a))v(q)/vertical bar x vertical bar(a). in R-N, (0,1) First, we establish Liouville-type theorems (i.e., non-existence of nontrivial nonnegative solutions) for system (0.1) via the method of scaling spheres. As an application, we derive the classification results for nonnegative solutions to system (0.1) when a = 0. Secondly, by exploiting the method of moving planes in integral forms, we prove the symmetry of the solutions to this nonlocal system. Finally, we study the existence and non-existence of positive least energy solution to system (0.1) via variational methods.