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    Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity
January  2019, 18(1): 519-538. doi: 10.3934/cpaa.2019026

Semiclassical states for fractional Choquard equations with critical growth

Hui Zhang 1,, , Jun Wang 2, and  Fubao Zhang 3,

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.
Keywords: Variational method, nonlocal nonlinearity, multiplicity of solutions, semiclassical state, critical growth.
Mathematics Subject Classification: Primary: 35J20, 35B33; Secondary: 35R11.
Citation: Hui Zhang, Jun Wang, Fubao Zhang. Semiclassical states for fractional Choquard equations with critical growth. Communications on Pure and 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.

[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.

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