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January  2019, 18(1): 519-538. doi: 10.3934/cpaa.2019026

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

* Corresponding author

Received  July 2017 Revised  October 2017 Published  August 2018

Fund Project: The work was supported by the National Natural Science Foundation of China (Nos. 11601204, 11671077, 11571140), Fellowship of Outstanding Young Scholars of Jiangsu Province (BK20160063), the Six big talent peaks project in Jiangsu Province (XYDXX-015), and Natural Science Foundation of Jiangsu Province (BK20150478)

In this paper, we are concerned with fractional Choquard equation
$ \ \ \ \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,$
where
$\epsilon>0$
is a parameter,
$0<α<1$
,
$0<μ<3$
,
$2_{μ,α}^* = \frac{6-μ}{3-2α}$
is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator,
$f$
is a continuous subcritical term, and
$F$
is the primitive function of
$f$
. By virtue of the method of Nehari manifold and Ljusternik-Schnirelmann category theory, we prove that the equation has a ground state for
$\epsilon$
small enough and investigate the relation between the number of solutions and the topology of the set where
$V$
attains its global minimum for small
$\epsilon$
. We also obtain sufficient conditions for the nonexistence of ground states.
Citation: Hui Zhang, Jun Wang, Fubao Zhang. Semiclassical states for fractional Choquard equations with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 519-538. doi: 10.3934/cpaa.2019026
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. Google Scholar

[2]

C. O. AlvesF. S. GaoM. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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S. CingolaniM. 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. Google Scholar

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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. Google Scholar

[9]

S. CingolaniS. 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. Google Scholar

[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. Google Scholar

[11]

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

[12]

E. Di NezzaG. 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. Google Scholar

[13]

R. L. Frank and E. Lenzmann, On ground states for the L2-critical boson star equation, arXiv: 0910.2721v2.Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/77), 93-105. Google Scholar

[18]

E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001.Google Scholar

[19]

P. L. Lions, The Choquard equation and related equations, Nonlinear Anal., 4 (1980), 1063-1073. doi: 10.1016/0362-546X(80)90016-4. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600. doi: 10.1007/BF02105068. Google Scholar

[27]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

Z. F. ShenF. 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. Google Scholar

[31]

A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010. Google Scholar

[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. Google Scholar

[33]

J. WangL. X. TianJ. 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. Google Scholar

[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. Google Scholar

[35]

H. ZhangJ. 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. Google Scholar

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. Google Scholar

[2]

C. O. AlvesF. S. GaoM. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

S. CingolaniM. 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. Google Scholar

[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. Google Scholar

[9]

S. CingolaniS. 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. Google Scholar

[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. Google Scholar

[11]

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

[12]

E. Di NezzaG. 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. Google Scholar

[13]

R. L. Frank and E. Lenzmann, On ground states for the L2-critical boson star equation, arXiv: 0910.2721v2.Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/77), 93-105. Google Scholar

[18]

E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001.Google Scholar

[19]

P. L. Lions, The Choquard equation and related equations, Nonlinear Anal., 4 (1980), 1063-1073. doi: 10.1016/0362-546X(80)90016-4. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600. doi: 10.1007/BF02105068. Google Scholar

[27]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

Z. F. ShenF. 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. Google Scholar

[31]

A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010. Google Scholar

[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. Google Scholar

[33]

J. WangL. X. TianJ. 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. Google Scholar

[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. Google Scholar

[35]

H. ZhangJ. 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. Google Scholar

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