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 R-+(n). 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).