-
Previous Article
Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator
- CPAA Home
- This Issue
-
Next Article
Estimates for sums of eigenvalues of the free plate via the fourier transform
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |
[1] |
Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020469 |
[2] |
Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020292 |
[3] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020450 |
[4] |
Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298 |
[5] |
Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 |
[6] |
Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260 |
[7] |
Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020388 |
[8] |
Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 |
[9] |
Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255 |
[10] |
Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 |
[11] |
Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020445 |
[12] |
Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297 |
[13] |
Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045 |
[14] |
Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020104 |
[15] |
Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029 |
[16] |
Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227 |
[17] |
Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125 |
[18] |
Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 |
[19] |
Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020443 |
[20] |
Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020354 |
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]