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January  2020, 19(1): 329-369. doi: 10.3934/cpaa.2020018

Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China

2. 

Universidade de Brasília, Departamento de Matemática, 70910-900, Brasília DF, Brazil

* Corresponding author

Received  December 2018 Revised  April 2019 Published  July 2019

Fund Project: The third author is supported by NSFC grant 11671364 and the Fourth author is supported by NSFC grant 11571317.

In this paper we are interested in the following critical coupled Hartree system
$ \left\{\begin{array}{l} (-\Delta)^{s} \tilde{u}+\lambda_{1}\tilde{u} = \alpha_{1}\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}, \ \ &&\mbox{in} \ \Omega,\\ (-\Delta)^{s} \tilde{v}+\lambda_{2}\tilde{v} = \alpha_{2}\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}, \ \ &&\mbox{in} \ \Omega,\\ \tilde{u} = \tilde{v} = 0, \ \ && \mbox{on} \ \partial \Omega, \end{array} \right. $
where
$ 0<s<1 $
,
$ \alpha_{1}, \alpha_{2}>0 $
,
$ \beta\neq0 $
,
$ 4s<\mu<N $
,
$ 2_{\mu}^{\ast} = (2N-\mu)/(N-2s) $
,
$ \Omega\subset\mathbb{R}^N(N\geq3) $
is a smooth bounded domain,
$ -\lambda_{1}(\Omega)<\lambda_{1}, \lambda_{2}<0 $
with
$ \lambda_{1}(\Omega) $
the first eigenvalue of
$ (-\Delta)^{s} $
under the Dirichlet boundary condition. Assume that the nonlinearity and the coupling terms are both of the upper critical growth due to the Hardy–Littlewood–Sobolev inequality, by applying the Dirichlet-to-Neumann map, we are able to obtain the existence of the ground state solution of the critical coupled Hartree system.
Citation: Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure and Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
References:
[1]

B. AbdellaouiV. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 34 (2009), 97-137.  doi: 10.1007/s00526-008-0177-2.

[2]

N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 26-61. 

[3]

C. O. AlvesD. CassaniC. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrodinger equation in $\mathbb{R}^{2}$, J. Differential Equations, 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021.

[4]

C. O. AlvesF. GaoM. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988.  doi: 10.1016/j.jde.2017.05.009.

[5]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020.

[6]

B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, A concave–convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, in press. doi: 10.1017/S0308210511000175.

[7]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 263 (2017), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.

[8]

T. BartschZ. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.

[9]

L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas., 7 (2000), 210-230. 

[10]

S. Bhattarai, On fractional Schrödinger systems of Choquard type, J. Differential Equations, 263 (2017), 3197-3229.  doi: 10.1016/j.jde.2017.04.034.

[11]

H. Brézis and T. Kato, Remarks on the Schrödinger operator with regular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. 

[12]

L. A. Caffarelli and L. Silvestre., An extension problem related to the fractional Laplacian, Commun. PDEs., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[13]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.

[14]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[15]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8.

[16]

Z. ChenC. Lin and W. Zou, Sign-changing solutions and phase separation for an elliptic system with critical exponent, Comm. Partial Differential Equations, 39 (2014), 1827-1859.  doi: 10.1080/03605302.2014.908391.

[17]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.  doi: 10.1007/s00526-014-0717-x.

[18]

F. DalfovoS. GiorginiL. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. 

[19]

E. Dancer and J. Wei, Spike Solutions in coupled nonlinear Schrödinger equations with Attractive Interaction, Tran. Amer. Math. Soc., 361 (2009), 1189-1208.  doi: 10.1090/S0002-9947-08-04735-1.

[20]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.

[21]

F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math., 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.

[22]

F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun.Contemp. Math., 20 (2018), no.4, 1750037, 22 pp. doi: 10.1142/S0219199717500377.

[23]

Z. GuoS. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706.  doi: 10.1016/j.jmaa.2016.08.069.

[24]

N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480.  doi: 10.1007/s00526-010-0347-x.

[25]

N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 180, Springer, 1972.

[26]

E. Lenzmann, Uniqueness of ground states for Hardy–Littlewood–Sobolev Hartree equations, Anal. PDE., 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[27]

M. Lewin and E. Lenzmann, On singularity formation for the $L^{2}$–critical boson star equation, Nonlinearity, 24 (2011), 3515-3540.  doi: 10.1088/0951-7715/24/12/009.

[28]

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

[29]

T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéare., 221 (2005), 403-439.  doi: 10.1016/j.anihpc.2004.03.004.

[30]

T. Lin and J. Wei, Ground state of N-coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, $n\leq3$, Commun. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.

[31]

T. Lin and J. Wei, Spikes in two-component systems of Nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 299 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011.

[32]

T. Lin and J. Wei, Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations, Nonlinearty, 19 (2006), 2755-2773.  doi: 10.1088/0951-7715/19/12/002.

[33]

A. Litvak, Self-focusing of powerful light beams by thermal effects, JETP Lett., 4 (1966), 230-232. 

[34]

Z. Liu and Z. Wang, Multiple bound states of nonlinear Schrödinger systems, Commun. Math. Phys., 282 (2008), 721–731. doi: 10.1007/s00220-008-0546-x.

[35]

H. Luo, Ground state solutions of Pohožaev type and Nehari type for a class of nonlinear Choquard equations, J. Math. Anal. Appl., 467 (2018), 842–862. doi: 10.1016/j.jmaa.2018.07.055.

[36]

L. MaiaE. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002.

[37]

C. Menyuk, Nonlinear pulse propagation in birefringence optical fiber, IEEE J. Quantum Electron, 23 (1987), 174-176. 

[38]

C. Menyuk, Pulse propagation in an elliptically birefringent Kerr medium, IEEE J. Quantum Electron, 25 (1989), 2674-2682. 

[39]

E. MontefuscoB. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.  doi: 10.4171/JEMS/103.

[40]

V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equation, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.

[41]

D. Mugnai, Pseudo–relativistic Hartree equation with general nonlinearity: existence, non-existence and variational identities, Adv. Nonlinear Stud., 13 (2013), 799-823.  doi: 10.1515/ans-2013-0403.

[42]

S. Peng, Y. Peng and Z. Wang, On elliptic systems with Sobolev critical growth, Calc. Var. Partial Differential Equations, 55 (2016), Art. 142, 30 pp. doi: 10.1007/s00526-016-1091-7.

[43]

S. PengW. Shuai and Q. Wang, Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, J. Differential Equations, 263 (2017), 709-731.  doi: 10.1016/j.jde.2017.02.053.

[44]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.

[45]

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.

[46]

Z. ShenF. Gao and M. Yang, Groundstates for nonlinear fractional Choquard equations with general nonlinearities, Math. Meth. Appl. Sci., 14 (2016), 4082-4098.  doi: 10.1002/mma.3849.

[47]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schröinger equations in $\mathbb{R}^{N}$, Commun. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.

[48] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, 1970. 
[49]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.  doi: 10.1007/s00526-010-0378-3.

[50]

J. Wang and J. Shi, Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction, Calc. Var. Partial Differential Equations, 56 (2017), Art. 168, 36 pp. doi: 10.1007/s00526-017-1268-8.

[51]

J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schröinger equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.

[52]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[53]

M. YangY. Wei and Y. Ding, Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities, Z. Angew. Math. Phys., 65 (2014), 41-68.  doi: 10.1007/s00033-013-0317-1.

[54]

Y. Zheng, Z. Shen and M. Yang, On critical pseudo-relativistic Hartree equation with potential well, Preprint.

show all references

References:
[1]

B. AbdellaouiV. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 34 (2009), 97-137.  doi: 10.1007/s00526-008-0177-2.

[2]

N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 26-61. 

[3]

C. O. AlvesD. CassaniC. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrodinger equation in $\mathbb{R}^{2}$, J. Differential Equations, 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021.

[4]

C. O. AlvesF. GaoM. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988.  doi: 10.1016/j.jde.2017.05.009.

[5]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020.

[6]

B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, A concave–convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, in press. doi: 10.1017/S0308210511000175.

[7]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 263 (2017), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.

[8]

T. BartschZ. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.

[9]

L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas., 7 (2000), 210-230. 

[10]

S. Bhattarai, On fractional Schrödinger systems of Choquard type, J. Differential Equations, 263 (2017), 3197-3229.  doi: 10.1016/j.jde.2017.04.034.

[11]

H. Brézis and T. Kato, Remarks on the Schrödinger operator with regular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. 

[12]

L. A. Caffarelli and L. Silvestre., An extension problem related to the fractional Laplacian, Commun. PDEs., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[13]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.

[14]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[15]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8.

[16]

Z. ChenC. Lin and W. Zou, Sign-changing solutions and phase separation for an elliptic system with critical exponent, Comm. Partial Differential Equations, 39 (2014), 1827-1859.  doi: 10.1080/03605302.2014.908391.

[17]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.  doi: 10.1007/s00526-014-0717-x.

[18]

F. DalfovoS. GiorginiL. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. 

[19]

E. Dancer and J. Wei, Spike Solutions in coupled nonlinear Schrödinger equations with Attractive Interaction, Tran. Amer. Math. Soc., 361 (2009), 1189-1208.  doi: 10.1090/S0002-9947-08-04735-1.

[20]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.

[21]

F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math., 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.

[22]

F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun.Contemp. Math., 20 (2018), no.4, 1750037, 22 pp. doi: 10.1142/S0219199717500377.

[23]

Z. GuoS. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706.  doi: 10.1016/j.jmaa.2016.08.069.

[24]

N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480.  doi: 10.1007/s00526-010-0347-x.

[25]

N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 180, Springer, 1972.

[26]

E. Lenzmann, Uniqueness of ground states for Hardy–Littlewood–Sobolev Hartree equations, Anal. PDE., 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[27]

M. Lewin and E. Lenzmann, On singularity formation for the $L^{2}$–critical boson star equation, Nonlinearity, 24 (2011), 3515-3540.  doi: 10.1088/0951-7715/24/12/009.

[28]

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

[29]

T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéare., 221 (2005), 403-439.  doi: 10.1016/j.anihpc.2004.03.004.

[30]

T. Lin and J. Wei, Ground state of N-coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, $n\leq3$, Commun. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.

[31]

T. Lin and J. Wei, Spikes in two-component systems of Nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 299 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011.

[32]

T. Lin and J. Wei, Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations, Nonlinearty, 19 (2006), 2755-2773.  doi: 10.1088/0951-7715/19/12/002.

[33]

A. Litvak, Self-focusing of powerful light beams by thermal effects, JETP Lett., 4 (1966), 230-232. 

[34]

Z. Liu and Z. Wang, Multiple bound states of nonlinear Schrödinger systems, Commun. Math. Phys., 282 (2008), 721–731. doi: 10.1007/s00220-008-0546-x.

[35]

H. Luo, Ground state solutions of Pohožaev type and Nehari type for a class of nonlinear Choquard equations, J. Math. Anal. Appl., 467 (2018), 842–862. doi: 10.1016/j.jmaa.2018.07.055.

[36]

L. MaiaE. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002.

[37]

C. Menyuk, Nonlinear pulse propagation in birefringence optical fiber, IEEE J. Quantum Electron, 23 (1987), 174-176. 

[38]

C. Menyuk, Pulse propagation in an elliptically birefringent Kerr medium, IEEE J. Quantum Electron, 25 (1989), 2674-2682. 

[39]

E. MontefuscoB. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.  doi: 10.4171/JEMS/103.

[40]

V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equation, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.

[41]

D. Mugnai, Pseudo–relativistic Hartree equation with general nonlinearity: existence, non-existence and variational identities, Adv. Nonlinear Stud., 13 (2013), 799-823.  doi: 10.1515/ans-2013-0403.

[42]

S. Peng, Y. Peng and Z. Wang, On elliptic systems with Sobolev critical growth, Calc. Var. Partial Differential Equations, 55 (2016), Art. 142, 30 pp. doi: 10.1007/s00526-016-1091-7.

[43]

S. PengW. Shuai and Q. Wang, Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, J. Differential Equations, 263 (2017), 709-731.  doi: 10.1016/j.jde.2017.02.053.

[44]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.

[45]

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.

[46]

Z. ShenF. Gao and M. Yang, Groundstates for nonlinear fractional Choquard equations with general nonlinearities, Math. Meth. Appl. Sci., 14 (2016), 4082-4098.  doi: 10.1002/mma.3849.

[47]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schröinger equations in $\mathbb{R}^{N}$, Commun. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.

[48] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, 1970. 
[49]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.  doi: 10.1007/s00526-010-0378-3.

[50]

J. Wang and J. Shi, Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction, Calc. Var. Partial Differential Equations, 56 (2017), Art. 168, 36 pp. doi: 10.1007/s00526-017-1268-8.

[51]

J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schröinger equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.

[52]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[53]

M. YangY. Wei and Y. Ding, Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities, Z. Angew. Math. Phys., 65 (2014), 41-68.  doi: 10.1007/s00033-013-0317-1.

[54]

Y. Zheng, Z. Shen and M. Yang, On critical pseudo-relativistic Hartree equation with potential well, Preprint.

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