Axisymmetry of locally bounded solutions to an Euler-Lagrangesystem of the weighted Hardy-Littlewood-Sobolev inequality

Yutian Lei; Zhongxue Lü

doi:10.3934/dcds.2013.33.1987

Article Contents

2013, Volume 33, Issue 5: 1987-2005. Doi: 10.3934/dcds.2013.33.1987

This issue Previous Article Two problems related to prescribed curvature measures Next Article The diffusive logistic model with a free boundary and seasonal succession

Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality

Yutian Lei^1, and
Zhongxue Lü^2,

1.
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097
2.
School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116

Received: November 2011

Revised: March 2012

Published: May 2013

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Abstract

This paper is concerned with the symmetry results for the $2k$-order singular Lane-Emden type partial differential system $$ \left\{\begin{array}{ll} (-\Delta)^k(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^k(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x), \end{array} \right. $$ and the weighted Hardy-Littlewood-Sobolev type integral system $$ \left \{ \begin{array}{l} u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy\\ v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy. \end{array} \right. $$ Here $x \in R^n \setminus \{0\}$. We first establish the equivalence of this integral system and an fractional order partial differential system, which includes the $2k$-order PDE system above. For the integral system, we prove that the positive locally bounded solutions are symmetric and decreasing about some axis by means of the method of moving planes in integral forms introduced by Chen-Li-Ou. In addition, we also show that the integrable solutions are locally bounded. Thus, the equivalence implies the positive solutions of the PDE system, particularly including the higher integer-order PDE system, also have the corresponding properties.

Keywords:

Higher-order Lane-Emden system,
weighted Hardy-Littlewood-Sobolev inequality,
method of moving planes,
axisymmetry.

Mathematics Subject Classification: 31B30, 35J48, 45E10, 45G05.

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References

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