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Multi-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity
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 |
$ \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. $ |
$ 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) $ |
$ -\lambda_{1}(\Omega)<\lambda_{1}, \lambda_{2}<0 $ |
$ \lambda_{1}(\Omega) $ |
$ (-\Delta)^{s} $ |
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. |
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N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 26-61. Google Scholar |
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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. |
[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. |
[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. 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. |
[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. |
[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 |
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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. |
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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.
|
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L. A. Caffarelli and L. Silvestre.,
An extension problem related to the fractional Laplacian, Commun. PDEs., 32 (2007), 1245-1260.
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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. |
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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. |
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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. |
[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. |
[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. Dalfovo, S. Giorgini, L. 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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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. Dalfovo, S. Giorgini, L. 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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[54] |
Y. Zheng, Z. Shen and M. Yang, On critical pseudo-relativistic Hartree equation with potential well, Preprint. Google Scholar |
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