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January  2020, 19(1): 123-144. doi: 10.3934/cpaa.2020008

Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth

Hua Jin 1,, , Wenbin Liu 1, , Huixing Zhang 1, and  Jianjun Zhang 2,

1. 

College of Science, China University of Mining and Technology, Xuzhou 221116, China
2. 

College of Mathematica and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

* Corresponding author

Received  October 2018 Revised  May 2019 Published  July 2019

We are concerned with nonlinear fractional Choquard equations involving critical growth in the sense of the Hardy-Littlewood-Sobolev inequality. Without the Ambrosetti-Rabinowitz condition or monotonicity condition on the nonlinearity, we establish the existence of radially symmetric ground state solutions.

Keywords: Fractional Choquard equations, ground states, Hardy-Littlewood-Sobolev critical growth.
Mathematics Subject Classification: Primary: 35B33, 35J60, 35Q55.
Citation: Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure and Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008
References:
[1]

G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tenstion effect, Arch. Ration. Mech. Anal., 144 (1998), 1-46.  doi: 10.1007/s002050050111.

[2]

H. Berestycki and P. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1990), 90-117.  doi: 10.1007/BF00250555.

[3]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[4]

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.

[5]

D. Cassani and J. Zhang, Choquard-type equations with Hardy-Littlewood-Sobolev upper critical growth, Adv. Nonlinear Anal., 8 (2019), 1184–1212. doi: 10.1515/anona-2018-0019.

[6]

X. J. Chang and Z. Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479.

[7]

Y. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.  doi: 10.1088/0951-7715/29/6/1827.

[8]

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

[9]

A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.

[10]

B. Calista, Regularity of solutions to the fractional Laplace equation, http://math.uchicago.edu/may/REU2014/REUPapers/Bernard.pdf, 2014.

[11]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchiker's guide to the fractional Sobolev space, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[12]

P. D'avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.

[13]

H. Genev and G. Venkov, Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 903-923.  doi: 10.3934/dcdss.2012.5.903.

[14]

Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 593 (2005), 385-424.  doi: 10.1142/S021949370500150X.

[15]

F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.

[16]

N. Laskin, Fractional quantum mechanics ans Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[17]

G. D. Li and C. L. Tang, Existence of ground state solutions for Choquard Equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear, Commun. Pure Appl. Anal., 18 (2019), 285-300.  doi: 10.3934/cpaa.2019015.

[18]

E. H. Lieb and M. Loss, Analysis, 2nd edn, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.

[19]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.

[20]

P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Analysis, 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6.

[21]

X. Q. Liu, J. Q. Liu and Z.-Q. Wang, Ground states for quasilinear Schrödinger equationns with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669.  doi: 10.1007/s00526-012-0497-0.

[22]

J. Liu, J. F. Liao and C. L. Tang, Ground state solution for a class of Schrödinger equations involving general critical growth term, Nonlinearity, 30 (2017), 899-911.  doi: 10.1088/1361-6544/aa5659.

[23]

P. Ma and J. Zhang, Existence and multiplicity of solutions for fractional Choquard equations, Nonlinear Analysis, 164 (2017), 100-117.  doi: 10.1016/j.na.2017.07.011.

[24]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.

[25]

V. Moroz and J. Van Schaftingen, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.

[26]

V. Moroz and J. Van Schaftingen, Existence of ground states for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.

[27]

V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theorey Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.

[28]

T. Mukherjee and K. Sreenadh, Fractional Choquard equations with critical nonlinearities, NoDEA Nonlinear Differential Equations Appl., 24 (2017). doi: 10.1007/s00030-017-0487-1.

[29]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[30]

Z. Shen, F. Gao and M. Yang, Ground states for nonlinear fractional Choquard equations with general nonlinearities, Math. Methods Appl. Sci., 39 (2016), 4082-4098.  doi: 10.1002/mma.3849.

[31]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace ooperator, Comm. Pure Apple. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[32]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transtion and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[33]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No, 30, Princeton University Press, Princeton, 1970.

[34]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016). doi: 10.1007/s00526-016-1045-0.

[35]

J. J. Zhang and W. M. Zou, A Berestycki-Lions Theorem revisited, Commun. Contemp. Math., 14 (2012), 1250033. doi: 10.1142/S0219199712500332.

show all references

References:
[1]

G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tenstion effect, Arch. Ration. Mech. Anal., 144 (1998), 1-46.  doi: 10.1007/s002050050111.

[2]

H. Berestycki and P. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1990), 90-117.  doi: 10.1007/BF00250555.

[3]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[4]

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.

[5]

D. Cassani and J. Zhang, Choquard-type equations with Hardy-Littlewood-Sobolev upper critical growth, Adv. Nonlinear Anal., 8 (2019), 1184–1212. doi: 10.1515/anona-2018-0019.

[6]

X. J. Chang and Z. Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479.

[7]

Y. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.  doi: 10.1088/0951-7715/29/6/1827.

[8]

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

[9]

A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.  doi: 10.1016/j.jmaa.2004.03.034.

[10]

B. Calista, Regularity of solutions to the fractional Laplace equation, http://math.uchicago.edu/may/REU2014/REUPapers/Bernard.pdf, 2014.

[11]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchiker's guide to the fractional Sobolev space, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[12]

P. D'avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.

[13]

H. Genev and G. Venkov, Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 903-923.  doi: 10.3934/dcdss.2012.5.903.

[14]

Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 593 (2005), 385-424.  doi: 10.1142/S021949370500150X.

[15]

F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.

[16]

N. Laskin, Fractional quantum mechanics ans Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[17]

G. D. Li and C. L. Tang, Existence of ground state solutions for Choquard Equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear, Commun. Pure Appl. Anal., 18 (2019), 285-300.  doi: 10.3934/cpaa.2019015.

[18]

E. H. Lieb and M. Loss, Analysis, 2nd edn, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.

[19]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.  doi: 10.1002/sapm197757293.

[20]

P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Analysis, 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6.

[21]

X. Q. Liu, J. Q. Liu and Z.-Q. Wang, Ground states for quasilinear Schrödinger equationns with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669.  doi: 10.1007/s00526-012-0497-0.

[22]

J. Liu, J. F. Liao and C. L. Tang, Ground state solution for a class of Schrödinger equations involving general critical growth term, Nonlinearity, 30 (2017), 899-911.  doi: 10.1088/1361-6544/aa5659.

[23]

P. Ma and J. Zhang, Existence and multiplicity of solutions for fractional Choquard equations, Nonlinear Analysis, 164 (2017), 100-117.  doi: 10.1016/j.na.2017.07.011.

[24]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.

[25]

V. Moroz and J. Van Schaftingen, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.

[26]

V. Moroz and J. Van Schaftingen, Existence of ground states for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.

[27]

V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theorey Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1.

[28]

T. Mukherjee and K. Sreenadh, Fractional Choquard equations with critical nonlinearities, NoDEA Nonlinear Differential Equations Appl., 24 (2017). doi: 10.1007/s00030-017-0487-1.

[29]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[30]

Z. Shen, F. Gao and M. Yang, Ground states for nonlinear fractional Choquard equations with general nonlinearities, Math. Methods Appl. Sci., 39 (2016), 4082-4098.  doi: 10.1002/mma.3849.

[31]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace ooperator, Comm. Pure Apple. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[32]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transtion and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[33]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No, 30, Princeton University Press, Princeton, 1970.

[34]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016). doi: 10.1007/s00526-016-1045-0.

[35]

J. J. Zhang and W. M. Zou, A Berestycki-Lions Theorem revisited, Commun. Contemp. Math., 14 (2012), 1250033. doi: 10.1142/S0219199712500332.

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