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

  • * Corresponding author: Zhao Liu

    * Corresponding author: Zhao Liu 

The first two authors were partly supported by the NNSF of China (No. 11371056), the first author was also supported by the NNSF of China (No. 11501021), the third author was partly supported by a US NSF grant and a Simons Fellowship from the Simons Foundation

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  • 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).

    Mathematics Subject Classification: Primary: 35J48; Secondary: 35B06, 45G15.


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