Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space

In this paper, we investigate the Hardy-Sobolev type integral systems (1) with Dirichlet boundary conditions in a half space ℝⁿ₊. We use the method of moving planes in integral forms introduced by Chen, Li and Ou [12] to prove that each pair of integrable positive solutions (u,v) of the above integral system is rotationally symmetric about x_n-axis in both subcritical and critical cases n-t/p+1 + n-s/q+1 ≥ n - 2m (Theorem 1.1). We also derive the non-existence of nontrivial nonnegative solutions with finite energy in the subcritical case (Theorem 1.2). By slightly modifying our arguments for studying the integral system, we can prove by a similar but simpler way that the same conclusions also hold for a single integral equation of Hardy-Sobolev type in both critical and subcritical cases (Theorem 1.3).