# American Institute of Mathematical Sciences

September  2021, 14(9): 3085-3096. doi: 10.3934/dcdss.2021079

## Symmetry of positive solutions for systems of fractional Hartree equations

 School of Science, China University of Geosciences, Beijing 100083, China

Received  January 2020 Revised  November 2020 Published  September 2021 Early access  June 2021

Fund Project: This work is supported by National Natural Science Foundation of China 11601493, and partially supported by Fundamental Research Funds for Central Universities 2652018058

In this paper, we deal with a system of fractional Hartree equations. By means of a direct method of moving planes, the radial symmetry and monotonicity of positive solutions are presented.

Citation: Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3085-3096. doi: 10.3934/dcdss.2021079
##### References:
 [1] G. Alberti and G. Bellettini, A nonlocal anisotropicmodel for phase transitions I: The optimal profile problem, Math. Ann., 310 (1998), 527-560.  doi: 10.1007/s002080050159. [2] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [3] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the Unione Matematica Italiana. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. [4] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331. [5] L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [7] L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6. [8] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013. [9] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038. [10] W. Chen, C. Li and B. Ou, Classiffication of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [11] W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029. [12] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall CRC Financial Mathematics Series, Chapman & HallCRC, Boca Raton, FL, 2004. [13] W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete and Continuous Dynamical Systems, 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117. [14] W. Dai, Y. Fang and G. Qin, Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes, J. Differential Equations, 265 (2018), 2044-2063.  doi: 10.1016/j.jde.2018.04.026. [15] J. Fröhlich, B. L. G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys., 274 (2007), 1-30.  doi: 10.1007/s00220-007-0272-9. [16] J. Giacomoni, T. Mukherjee and K. Sreenadh, Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 311-337.  doi: 10.3934/dcdss.2019022. [17] D. Li, C. Miao and X. Zhang, The focusing energy-critical Hartree equation, J. Differential Equations, 246 (2009), 1139-1163. doi: 10.1016/j.jde.2008.05.013. [18] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845. [19] X. Liu, Symmetry of positive solutions for the fractional Hartree equation, Acta Math.Sci., 39 (2019), 1508-1516.  doi: 10.1007/s10473-019-0603-x. [20] C. Miao, G. Xu and and L. Zhao, Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., 114 (2009), 213-236.  doi: 10.4064/cm114-2-5. [21] S. Serfaty and J. L. V$\acute{a}$zquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Differential Equations, 49 (2014), 1091-1120.  doi: 10.1007/s00526-013-0613-9. [22] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468. [23] Z. Shen, Z. Han and Q. Zhang, Ground states of nonlinear Schrödinger equations with fractional Laplacians, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 2115-2125.  doi: 10.3934/dcdss.2019136. [24] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [25] J. Sun, T.-F. Wu and Z. Feng, Non-autonomous Schrödinger-Poisson system in $R^3$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933.  doi: 10.3934/dcds.2018077. [26] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. Se MA, 49 (2009), 33-44. [27] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in "Nonlinear Partial Differential Equations", Springer, Heidelberg, 2012, 271–298. doi: 10.1007/978-3-642-25361-4_15.

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##### References:
 [1] G. Alberti and G. Bellettini, A nonlocal anisotropicmodel for phase transitions I: The optimal profile problem, Math. Ann., 310 (1998), 527-560.  doi: 10.1007/s002080050159. [2] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [3] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the Unione Matematica Italiana. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3. [4] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331. [5] L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [7] L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6. [8] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013. [9] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038. [10] W. Chen, C. Li and B. Ou, Classiffication of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [11] W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029. [12] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall CRC Financial Mathematics Series, Chapman & HallCRC, Boca Raton, FL, 2004. [13] W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete and Continuous Dynamical Systems, 39 (2019), 1389-1403.  doi: 10.3934/dcds.2018117. [14] W. Dai, Y. Fang and G. Qin, Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes, J. Differential Equations, 265 (2018), 2044-2063.  doi: 10.1016/j.jde.2018.04.026. [15] J. Fröhlich, B. L. G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys., 274 (2007), 1-30.  doi: 10.1007/s00220-007-0272-9. [16] J. Giacomoni, T. Mukherjee and K. Sreenadh, Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 311-337.  doi: 10.3934/dcdss.2019022. [17] D. Li, C. Miao and X. Zhang, The focusing energy-critical Hartree equation, J. Differential Equations, 246 (2009), 1139-1163. doi: 10.1016/j.jde.2008.05.013. [18] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  doi: 10.1007/BF01609845. [19] X. Liu, Symmetry of positive solutions for the fractional Hartree equation, Acta Math.Sci., 39 (2019), 1508-1516.  doi: 10.1007/s10473-019-0603-x. [20] C. Miao, G. Xu and and L. Zhao, Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., 114 (2009), 213-236.  doi: 10.4064/cm114-2-5. [21] S. Serfaty and J. L. V$\acute{a}$zquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Differential Equations, 49 (2014), 1091-1120.  doi: 10.1007/s00526-013-0613-9. [22] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468. [23] Z. Shen, Z. Han and Q. Zhang, Ground states of nonlinear Schrödinger equations with fractional Laplacians, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 2115-2125.  doi: 10.3934/dcdss.2019136. [24] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [25] J. Sun, T.-F. Wu and Z. Feng, Non-autonomous Schrödinger-Poisson system in $R^3$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933.  doi: 10.3934/dcds.2018077. [26] E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. Se MA, 49 (2009), 33-44. [27] J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in "Nonlinear Partial Differential Equations", Springer, Heidelberg, 2012, 271–298. doi: 10.1007/978-3-642-25361-4_15.
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