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

Wei Dai; Zhao Liu; Guozhen Lu

doi:10.3934/cpaa.2017061

Article Contents

2017, Volume 16, Issue 4: 1253-1264. Doi: 10.3934/cpaa.2017061

This issue Previous Article Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect Next Article Stability of the composite wave for the inflow problem on the micropolar fluid model

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

1.
chool of Mathematics and Systems Science, Beihang University (BUAA), Beijing, 100191, China
2.
School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang, 330038, China
3.
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

^* Corresponding author: Zhao Liu
^* Corresponding author: Zhao Liu

Received: June 30, 2016

Revised: December 31, 2016

Published: May 2017

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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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).

Keywords:

Liouville type theorem,
Dirichlet problem,
half space,
method of moving planes in integral forms,
rotational symmetry,
nonexistence.

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

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References

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