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 & 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.  Google Scholar

[2]

C. BrandleE. ColoradoA. 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.  Google Scholar

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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.  Google Scholar

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L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

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L. A. CaffarelliS. 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.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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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.  Google Scholar

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W. ChenY. 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.  Google Scholar

[9]

W. ChenC. 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.  Google Scholar

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

[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.  Google Scholar

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall CRC Financial Mathematics Series, Chapman & HallCRC, Boca Raton, FL, 2004.  Google Scholar

[13]

W. DaiY. FangJ. HuangY. 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.  Google Scholar

[14]

W. DaiY. 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.  Google Scholar

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J. FröhlichB. 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.  Google Scholar

[16]

J. GiacomoniT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

C. MiaoG. 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.  Google Scholar

[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.  Google Scholar

[22]

J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[23]

Z. ShenZ. 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.  Google Scholar

[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.  Google Scholar

[25]

J. SunT.-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.  Google Scholar

[26]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. Se MA, 49 (2009), 33-44.   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

C. BrandleE. ColoradoA. 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.  Google Scholar

[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.  Google Scholar

[4]

L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

[5]

L. A. CaffarelliS. 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.  Google Scholar

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[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.  Google Scholar

[8]

W. ChenY. 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.  Google Scholar

[9]

W. ChenC. 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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall CRC Financial Mathematics Series, Chapman & HallCRC, Boca Raton, FL, 2004.  Google Scholar

[13]

W. DaiY. FangJ. HuangY. 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.  Google Scholar

[14]

W. DaiY. 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.  Google Scholar

[15]

J. FröhlichB. 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.  Google Scholar

[16]

J. GiacomoniT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

C. MiaoG. 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.  Google Scholar

[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.  Google Scholar

[22]

J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[23]

Z. ShenZ. 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.  Google Scholar

[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.  Google Scholar

[25]

J. SunT.-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.  Google Scholar

[26]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. Se MA, 49 (2009), 33-44.   Google Scholar

[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.  Google Scholar

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