# American Institute of Mathematical Sciences

• Previous Article
Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates
• CPAA Home
• This Issue
• Next Article
Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics
November  2016, 15(6): 2117-2134. doi: 10.3934/cpaa.2016030

## Some properties of positive solutions for an integral system with the double weighted Riesz potentials

 1 College of Sciences, Hunan Agriculture University, Changsha Hunan 410128, China 2 Institute of Applied Physics and Computational Mathematics, P.O.Box 8009-28, Beijing 100088 3 School of Mathematical Sciences, Xiamen University, Xiamen Fujian 361005, China

Received  December 2015 Revised  April 2016 Published  September 2016

In this paper, we study some important properties of positive solutions for a nonlinear integral system. With the help of the method of moving planes in an integral form, we show that under certain integrable conditions, all of positive solutions to this system are radially symmetric and decreasing with respect to the origin. Meanwhile, using the regularity lifting lemma, which was recently introduced by Chen and Li in [1], we obtain the optimal integrable intervals and sharp asymptotic behaviors for such positive solutions, which characterize the closeness of system to some extent.
Citation: Jiankai Xu, Song Jiang, Huoxiong Wu. Some properties of positive solutions for an integral system with the double weighted Riesz potentials. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2117-2134. doi: 10.3934/cpaa.2016030
##### References:
 [1] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Ser. Differ. Equ. Dyn. Syst., (2010).   Google Scholar [2] C. Jin and C. Li, Qualitative analysis of some systems of equations,, \emph{Calc. Var. Partial Differential Equations}, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar [3] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar [4] Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear system,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar [5] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations,, \emph{Calc. Var. Partial Differential Equations}, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar [6] Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 193.  doi: 10.3934/cpaa.2011.10.193.  Google Scholar [7] C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, \emph{Comm. Pure Appl. Anal.}, 6 (2007), 453.  doi: 10.3934/cpaa.2007.6.453.  Google Scholar [8] C. Li and L. Ma, Uniqueness of positive bound states to schrödinger systems with critical expoents,, \emph{SIAM J. Math. Anal.}, 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar [9] E. H. Lieb and M. Loss, Analysis,, 2$^{nd}$ edition, (2001).  doi: 10.1090/gsm/014.  Google Scholar [10] G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality,, \emph{Calc. Var. Partial Differential Equations}, 42 (2011), 563.  doi: 10.1007/s00526-011-0398-7.  Google Scholar [11] J. Xu, H. Wu and Z. Tan, Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations,, \emph{J. Math. Anal. Appl.}, 427 (2015), 307.  doi: 10.1016/j.jmaa.2015.02.043.  Google Scholar [12] Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system,, \emph{Nonlinear Anal.}, 75 (2012), 1989.  doi: 10.1016/j.na.2011.09.051.  Google Scholar

show all references

##### References:
 [1] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Ser. Differ. Equ. Dyn. Syst., (2010).   Google Scholar [2] C. Jin and C. Li, Qualitative analysis of some systems of equations,, \emph{Calc. Var. Partial Differential Equations}, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar [3] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar [4] Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear system,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar [5] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations,, \emph{Calc. Var. Partial Differential Equations}, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar [6] Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 193.  doi: 10.3934/cpaa.2011.10.193.  Google Scholar [7] C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, \emph{Comm. Pure Appl. Anal.}, 6 (2007), 453.  doi: 10.3934/cpaa.2007.6.453.  Google Scholar [8] C. Li and L. Ma, Uniqueness of positive bound states to schrödinger systems with critical expoents,, \emph{SIAM J. Math. Anal.}, 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar [9] E. H. Lieb and M. Loss, Analysis,, 2$^{nd}$ edition, (2001).  doi: 10.1090/gsm/014.  Google Scholar [10] G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality,, \emph{Calc. Var. Partial Differential Equations}, 42 (2011), 563.  doi: 10.1007/s00526-011-0398-7.  Google Scholar [11] J. Xu, H. Wu and Z. Tan, Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations,, \emph{J. Math. Anal. Appl.}, 427 (2015), 307.  doi: 10.1016/j.jmaa.2015.02.043.  Google Scholar [12] Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system,, \emph{Nonlinear Anal.}, 75 (2012), 1989.  doi: 10.1016/j.na.2011.09.051.  Google Scholar
 [1] Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987 [2] Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791 [3] Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164 [4] Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951 [5] Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018 [6] Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935 [7] Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057 [8] Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171 [9] Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027 [10] Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018 [11] Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223 [12] Martí Prats. Beltrami equations in the plane and Sobolev regularity. Communications on Pure & Applied Analysis, 2018, 17 (2) : 319-332. doi: 10.3934/cpaa.2018018 [13] Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653 [14] Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925 [15] Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121 [16] José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138 [17] Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015 [18] Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure & Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008 [19] Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1 [20] Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041

2019 Impact Factor: 1.105