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

Hua Jin; Wenbin Liu; Huixing Zhang; Jianjun Zhang

doi:10.3934/cpaa.2020008

Article Contents

2020, Volume 19, Issue 1: 123-144. Doi: 10.3934/cpaa.2020008

This issue Previous Article Estimates for sums of eigenvalues of the free plate via the fourier transform Next Article Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator

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
^* Corresponding author

Received: October 2018

Revised: May 2019

Published: July 2019

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Full Text(HTML)

Related Papers

Cited by

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.