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

Abstract

In this paper, we investigate the Hardy-Sobolev type integral systems (1) with Dirichlet boundary conditions in a half space $\mathbb{R}_+^n$. 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 $\frac{n-t}{p+1}+\frac{n-s}{q+1}\geq n-2m$ (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).