Liouville-type theorems for integral and PDE systems with critical exponents in a half space

Let be the n-dimensional upper half Euclidean space, and let m be a positive integer. In this paper, we consider the following integral system with critical exponents in whereis the Green’s function in with the Dirichlet boundary conditions. Then we use the method of moving planes in integral forms presented by Chen et al. to prove Liouville-type theorems. Under some mild growth conditions, we also establish the equivalence between IE system and corresponding PDE system.