-
Previous Article
Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities
- CPAA Home
- This Issue
-
Next Article
Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity
Semiclassical states for fractional Choquard equations with critical growth
1. | Department of Mathematics, Jinling Institute of Technology, Nanjing 211169, China |
2. | Department of Mathematics, Jiangsu University, Zhenjiang 212013, China |
3. | Department of Mathematics, Southeast University, Nanjing 210096, China |
$ \ \ \ \epsilon^{2α}(-Δ)^α u+V(x)u\\ = \epsilon^{μ-3}\Bigl(\int_{\mathbb{R}^3}\frac{|u(y)|^{2_{μ,α}^*}+F(u(y))}{|x-y|^μ}dy\Bigr)\Bigl(|u|^{2_{μ,α}^*-2}u+\frac{1}{2_{μ,α}^*}f(u)\Bigr)\ {\rm in}\ \mathbb{R}^3,$ |
$\epsilon>0$ |
$0<α<1$ |
$0<μ<3$ |
$2_{μ,α}^* = \frac{6-μ}{3-2α}$ |
$f$ |
$F$ |
$f$ |
$\epsilon$ |
$V$ |
$\epsilon$ |
References:
[1] |
C. O. Alves and G. M. Figueiredo,
On multiplicity and concentration of
positive solutions for a class of quasilinear problems with critical
exponential growth in $ \mathbb{R}^{N}$, J. Differential Equations, 246 (2009), 1288-1331.
doi: 10.1016/j.jde.2008.08.004. |
[2] |
C. O. Alves, F. S. Gao, M. Squassina and M. B. Yang,
Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988.
doi: 10.1016/j.jde.2017.05.009. |
[3] |
C. O. Alves and M. B. Yang, Multiplicity and concentration of solutions for a quasilinear Choquard equation, J. Math. Phys., 55 (2014), 061502, 21pp.
doi: 10.1063/1.4884301. |
[4] |
C. O. Alves and M. B. Yang,
Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations, 257 (2014), 4133-4164.
doi: 10.1016/j.jde.2014.08.004. |
[5] |
C. O. Alves and M. B. Yang,
Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 23-58.
doi: 10.1017/S0308210515000311. |
[6] |
Y. H. Chen and C. G. Liu,
Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.
doi: 10.1088/0951-7715/29/6/1827. |
[7] |
S. Cingolani, M. Clapp and S. Secchi,
Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[8] |
S. Cingolani and M. Lazzo,
Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.
doi: 10.1006/jdeq.1999.3662. |
[9] |
S. Cingolani, S. Secchi and M. Squassina,
Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.
doi: 10.1017/S0308210509000584. |
[10] |
A. Cotsiolis and N. 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. |
[11] |
P. d'Avenia, G. Siciliano and M. Squassina,
On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.
doi: 10.1142/S0218202515500384. |
[12] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. des Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[13] |
R. L. Frank and E. Lenzmann, On ground states for the L2-critical boson star equation, arXiv: 0910.2721v2. |
[14] |
F. S. Gao and M. B. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., DOI: 10.1007/s11425-016-9067-5. |
[15] |
X. M. He and W. M. Zou,
Existence and concentration behavior of
positive solutions for a Kirchhoff equation in $ \mathbb{R}^{3}$, J.Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[16] |
N. Laskin,
Fractional quantum mechanics and L'evy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[17] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/77), 93-105.
|
[18] |
E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001. |
[19] |
P. L. Lions,
The Choquard equation and related equations, Nonlinear Anal., 4 (1980), 1063-1073.
doi: 10.1016/0362-546X(80)90016-4. |
[20] |
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. |
[21] |
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. |
[22] |
V. Moroz and J. Van Schaftingen,
Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.
doi: 10.1007/s00526-014-0709-x. |
[23] |
V. Moroz and J. Van Schaftingen,
Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Maths. Soc., 367 (2015), 6557-6579.
doi: 10.1090/S0002-9947-2014-06289-2. |
[24] |
V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: HardyLittlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12 pp.
doi: 10.1142/S0219199715500054. |
[25] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calculus Var. Partial Differ. Equ., 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[26] |
R. Penrose,
On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.
doi: 10.1007/BF02105068. |
[27] |
P. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[28] |
S. Secchi,
Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R}^3$, J. Math. Phys., 54 (2013), 031501.
doi: 10.1063/1.4793990. |
[29] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[30] |
Z. F. Shen, F. S. Gao and M. B. 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] |
A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010. |
[32] |
J. Wang and J. P. Shi,
Standing waves of a weakly coupled Schrödinger system with distinct potential functions, J. Differential Equations, 260 (2016), 1830-1864.
doi: 10.1016/j.jde.2015.09.052. |
[33] |
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang,
Multiplicity and
concentration of positive solutions for a Kirchhoff type problem
with critical growth, J. Differential Equations, 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[34] |
H. Weitzner and G. M. Zaslavsky,
Some applications of fractional equations, Chaotic transport and complexity in classical and quantum dynamics, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281.
doi: 10.1016/S1007-5704(03)00049-2. |
[35] |
H. Zhang, J. X. Xu and F. B. Zhang,
Existence and multiplicity of solutions for a generalized Choquard equation, Comput. Math. Appl., 73 (2017), 1803-1814.
doi: 10.1016/j.camwa.2017.02.026. |
show all references
References:
[1] |
C. O. Alves and G. M. Figueiredo,
On multiplicity and concentration of
positive solutions for a class of quasilinear problems with critical
exponential growth in $ \mathbb{R}^{N}$, J. Differential Equations, 246 (2009), 1288-1331.
doi: 10.1016/j.jde.2008.08.004. |
[2] |
C. O. Alves, F. S. Gao, M. Squassina and M. B. Yang,
Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988.
doi: 10.1016/j.jde.2017.05.009. |
[3] |
C. O. Alves and M. B. Yang, Multiplicity and concentration of solutions for a quasilinear Choquard equation, J. Math. Phys., 55 (2014), 061502, 21pp.
doi: 10.1063/1.4884301. |
[4] |
C. O. Alves and M. B. Yang,
Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations, 257 (2014), 4133-4164.
doi: 10.1016/j.jde.2014.08.004. |
[5] |
C. O. Alves and M. B. Yang,
Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 23-58.
doi: 10.1017/S0308210515000311. |
[6] |
Y. H. Chen and C. G. Liu,
Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.
doi: 10.1088/0951-7715/29/6/1827. |
[7] |
S. Cingolani, M. Clapp and S. Secchi,
Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[8] |
S. Cingolani and M. Lazzo,
Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.
doi: 10.1006/jdeq.1999.3662. |
[9] |
S. Cingolani, S. Secchi and M. Squassina,
Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.
doi: 10.1017/S0308210509000584. |
[10] |
A. Cotsiolis and N. 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. |
[11] |
P. d'Avenia, G. Siciliano and M. Squassina,
On fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.
doi: 10.1142/S0218202515500384. |
[12] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. des Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[13] |
R. L. Frank and E. Lenzmann, On ground states for the L2-critical boson star equation, arXiv: 0910.2721v2. |
[14] |
F. S. Gao and M. B. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., DOI: 10.1007/s11425-016-9067-5. |
[15] |
X. M. He and W. M. Zou,
Existence and concentration behavior of
positive solutions for a Kirchhoff equation in $ \mathbb{R}^{3}$, J.Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[16] |
N. Laskin,
Fractional quantum mechanics and L'evy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[17] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/77), 93-105.
|
[18] |
E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001. |
[19] |
P. L. Lions,
The Choquard equation and related equations, Nonlinear Anal., 4 (1980), 1063-1073.
doi: 10.1016/0362-546X(80)90016-4. |
[20] |
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. |
[21] |
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. |
[22] |
V. Moroz and J. Van Schaftingen,
Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.
doi: 10.1007/s00526-014-0709-x. |
[23] |
V. Moroz and J. Van Schaftingen,
Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Maths. Soc., 367 (2015), 6557-6579.
doi: 10.1090/S0002-9947-2014-06289-2. |
[24] |
V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: HardyLittlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12 pp.
doi: 10.1142/S0219199715500054. |
[25] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calculus Var. Partial Differ. Equ., 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[26] |
R. Penrose,
On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.
doi: 10.1007/BF02105068. |
[27] |
P. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[28] |
S. Secchi,
Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R}^3$, J. Math. Phys., 54 (2013), 031501.
doi: 10.1063/1.4793990. |
[29] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[30] |
Z. F. Shen, F. S. Gao and M. B. 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] |
A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010. |
[32] |
J. Wang and J. P. Shi,
Standing waves of a weakly coupled Schrödinger system with distinct potential functions, J. Differential Equations, 260 (2016), 1830-1864.
doi: 10.1016/j.jde.2015.09.052. |
[33] |
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang,
Multiplicity and
concentration of positive solutions for a Kirchhoff type problem
with critical growth, J. Differential Equations, 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[34] |
H. Weitzner and G. M. Zaslavsky,
Some applications of fractional equations, Chaotic transport and complexity in classical and quantum dynamics, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281.
doi: 10.1016/S1007-5704(03)00049-2. |
[35] |
H. Zhang, J. X. Xu and F. B. Zhang,
Existence and multiplicity of solutions for a generalized Choquard equation, Comput. Math. Appl., 73 (2017), 1803-1814.
doi: 10.1016/j.camwa.2017.02.026. |
[1] |
Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2065-2100. doi: 10.3934/cpaa.2021058 |
[2] |
Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292 |
[3] |
Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity. Electronic Research Archive, 2021, 29 (5) : 3281-3295. doi: 10.3934/era.2021038 |
[4] |
Marcos L. M. Carvalho, José Valdo A. Goncalves, Claudiney Goulart, Olímpio H. Miyagaki. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth. Communications on Pure and Applied Analysis, 2019, 18 (1) : 83-106. doi: 10.3934/cpaa.2019006 |
[5] |
Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure and Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99 |
[6] |
Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure and Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417 |
[7] |
Claudianor O. Alves, J. V. Gonçalves, Olimpio Hiroshi Miyagaki. Remarks on multiplicity of positive solutions of nonlinear elliptic equations in $IR^N$ with critical growth. Conference Publications, 1998, 1998 (Special) : 51-57. doi: 10.3934/proc.1998.1998.51 |
[8] |
Claudianor O. Alves, Giovany M. Figueiredo, Gaetano Siciliano. Ground state solutions for fractional scalar field equations under a general critical nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2199-2215. doi: 10.3934/cpaa.2019099 |
[9] |
Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025 |
[10] |
Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235 |
[11] |
Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054 |
[12] |
Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 |
[13] |
Junping Shi, Ratnasingham Shivaji. Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 559-571. doi: 10.3934/dcds.2001.7.559 |
[14] |
Xiaohui Yu. Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros. Communications on Pure and Applied Analysis, 2013, 12 (1) : 451-459. doi: 10.3934/cpaa.2013.12.451 |
[15] |
Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 |
[16] |
Kyril Tintarev. Positive solutions of elliptic equations with a critical oscillatory nonlinearity. Conference Publications, 2007, 2007 (Special) : 974-981. doi: 10.3934/proc.2007.2007.974 |
[17] |
Monica Musso, Donato Passaseo. Multiple solutions of Neumann elliptic problems with critical nonlinearity. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 301-320. doi: 10.3934/dcds.1999.5.301 |
[18] |
Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079 |
[19] |
Gonzalo Galiano, Sergey Shmarev, Julian Velasco. Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1479-1501. doi: 10.3934/dcds.2015.35.1479 |
[20] |
Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]