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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

Yu Zheng 1, , Carlos A. Santos 2, , Zifei Shen 1, and  Minbo Yang 1,,

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.
Keywords: Hartree system, Brezis-Nirenberg problem, Hardy-Littlewood-Sobolev inequality, critical exponent.
Mathematics Subject Classification: 35J25, 35J60, 35A15.
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 & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
References:
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B. Abdellaoui, V. 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. Google Scholar

[2]

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

[3]

C. O. Alves, D. Cassani, C. 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. Google Scholar

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

[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. Google Scholar

[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. Google Scholar

[7]

B. Barrios, E. Colorado, A. 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. Google Scholar

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T. Bartsch, Z. 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. Google Scholar

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

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

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

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

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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Z. Chen, C. 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. Google Scholar

[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. Google Scholar

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F. Dalfovo, S. Giorgini, L. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

Z. Guo, S. 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. Google Scholar

[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. Google Scholar

[25]

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

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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. Google Scholar

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[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. Google Scholar

[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. Google Scholar

[36]

L. Maia, E. 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. Google Scholar

[37]

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

[38]

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

[39]

E. Montefusco, B. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[43]

S. Peng, W. 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. Google Scholar

[44]

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

[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. Google Scholar

[46]

Z. Shen, F. 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. Google Scholar

[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. Google Scholar

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[53]

M. Yang, Y. 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. Google Scholar

[54]

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

show all references

References:
[1]

B. Abdellaoui, V. 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. Google Scholar

[2]

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

[3]

C. O. Alves, D. Cassani, C. 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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[7]

B. Barrios, E. Colorado, A. 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. Google Scholar

[8]

T. Bartsch, Z. 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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

Z. Chen, C. 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. Google Scholar

[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. Google Scholar

[18]

F. Dalfovo, S. Giorgini, L. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

Z. Guo, S. 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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

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

[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. Google Scholar

[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. Google Scholar

[36]

L. Maia, E. 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. Google Scholar

[37]

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

[38]

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

[39]

E. Montefusco, B. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[43]

S. Peng, W. 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. Google Scholar

[44]

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

[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. Google Scholar

[46]

Z. Shen, F. 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. Google Scholar

[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. Google Scholar

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[53]

M. Yang, Y. 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. Google Scholar

[54]

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

[1]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951
[2]

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