July  2016, 36(7): 3791-3810. doi: 10.3934/dcds.2016.36.3791

Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system

1. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

2. 

School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, 221116

Received  May 2015 Revised  November 2015 Published  March 2016

This paper is concerned with the properties of solutions for the weighted Hardy-Littlewood-Sobolev type integral system \begin{equation} \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.                                                                              (1) \end{equation} and the fractional order partial differential system \begin{equation} \label{PDE} \left\{\begin{array}{ll} (-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x). \end{array}                                                                       (2) \right. \end{equation} Here $x \in R^n \setminus \{0\}$. Due to $0 < p, q < \infty$, we need more complicated analytical techniques to handle the case $0< p <1$ or $0< q <1$. We first establish the equivalence of integral system (1) and fractional order partial differential system (2). For integral system (1), we prove that the integrable solutions are locally bounded. In addition, we also show 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. Thus, the equivalence implies the positive solutions of the PDE system, also have the corresponding properties. This paper extends previous results obtained by other authors to the general case.
Citation: Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791
References:
[1]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962. doi: 10.1090/S0002-9939-07-09232-5.

[2]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12(6) (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.

[3]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1977.

[5]

F. Hang, On the integral systems related to HLS inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.

[6]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X.

[7]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5.

[8]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system,, Calc. Var. Partial Differential Equations., ().  doi: 10.1007/s00526-011-0450-7.

[9]

Y. Lei and Z. Lü, Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 30 (2014), 547-558. doi: 10.3934/dcds.2013.33.1987.

[10]

Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Comm. Pure Appl. Anal., 10 (2011), 193-207. doi: 10.3934/cpaa.2011.10.193.

[11]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6, (2007), 453-464. doi: 10.3934/cpaa.2007.6.453.

[12]

M. Onodera, On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510. doi: 10.1016/j.jmaa.2011.12.004.

[13]

E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.

[14]

J. Wei and X. Xu, Classification of solutions of highter order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.

show all references

References:
[1]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136 (2008), 955-962. doi: 10.1090/S0002-9939-07-09232-5.

[2]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12(6) (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.

[3]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1977.

[5]

F. Hang, On the integral systems related to HLS inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.

[6]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X.

[7]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5.

[8]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system,, Calc. Var. Partial Differential Equations., ().  doi: 10.1007/s00526-011-0450-7.

[9]

Y. Lei and Z. Lü, Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality, Discrete Contin. Dyn. Syst., 30 (2014), 547-558. doi: 10.3934/dcds.2013.33.1987.

[10]

Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Comm. Pure Appl. Anal., 10 (2011), 193-207. doi: 10.3934/cpaa.2011.10.193.

[11]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6, (2007), 453-464. doi: 10.3934/cpaa.2007.6.453.

[12]

M. Onodera, On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510. doi: 10.1016/j.jmaa.2011.12.004.

[13]

E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.

[14]

J. Wei and X. Xu, Classification of solutions of highter order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.

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