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

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

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.

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 & Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008
References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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W. X. ChenC. 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.  Google Scholar

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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.  Google Scholar

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

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E. Di NezzaG. 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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[21]

X. Q. LiuJ. 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.  Google Scholar

[22]

J. LiuJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

Z. ShenF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. X. ChenC. 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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

E. Di NezzaG. 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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

X. Q. LiuJ. 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.  Google Scholar

[22]

J. LiuJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

Z. ShenF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[35]

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

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