American Institute of Mathematical Sciences

Advanced
August  2019, 39(8): 4565-4612. doi: 10.3934/dcds.2019188

On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

Van Duong Dinh 1,2, and  Binhua Feng 3,,

1. 

Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France
2. 

Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam
3. 

Department of Mathematics, Northwest Normal University, Lanzhou, China

* Corresponding author: Binhua Feng

Received  September 2018 Revised  February 2019 Published  May 2019

We undertake a comprehensive study for the fractional nonlinear Schrödinger equation
$ i\partial_t u - (-\Delta)^s u = \mu_1 |u|^{\alpha_1} u + \mu_2 |u|^{\alpha_2} u, \quad u(0) = u_0, $
where
$ \frac{d}{2d-1} \leq s <1 $
,
$ 0 < \alpha_1 <\alpha_2 < \frac{4s}{d-2s} $
. Firstly, we establish the local and global well-posedness results for non-radial and radial
$ H^s $
 initial data, radial
$ \dot{H}^{s_c}\cap \dot{H}^s $
 initial data, where
$ s_c = \frac{d}{2}-\frac{2s}{\alpha_2} $
. Secondly, we study the asymptotic behavior of global radial
$ H^s $
 solutions. Of particular interest is the
$ L^2 $
-critical case and the results in this case are conditional on a conjectured global existence and spacetime estimate for the
$ L^2 $
-critical fractional nonlinear Schrödinger equation. Thirdly, we obtain sufficient conditions about existence of blow-up radial
$ \dot{H}^{s_c} \cap \dot{H}^s $
 solutions, and derive the sharp threshold mass of blow-up and global existence for this equation with
$ L^2 $
-critical and
$ L^2 $
-subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blow-up solutions in both
$ L^2 $
-critical and
$ L^2 $
-supercritical cases, including mass-concentration and limiting profile.
Keywords: Fractional nonlinear Schrödinger equation, global existence, scattering, blow-up criteria, blow-up dynamics.
Mathematics Subject Classification: Primary: 35B44; Secondary: 35Q55.
Citation: Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188
References:
[1]

J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976. Google Scholar

[2]

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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T. Boulenger, D. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603. doi: 10.1016/j.jfa.2016.08.011. Google Scholar

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T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A. Google Scholar

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T. Cazenave, Semilinear Schrödinger Equations, Courant Lectures Notes in Mathematics 10, New York, AMS, 2003. doi: 10.1090/cln/010. Google Scholar

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X. Cheng, C. Miao and L. Zhao, Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934. doi: 10.1016/j.jde.2016.04.031. Google Scholar

[7]

Y. Cho, Short-range scattering of Hartree type fractional NLS, J. Differential Equations, 262 (2017), 116-144. doi: 10.1016/j.jde.2016.09.025. Google Scholar

[8]

Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equations with Hartree type nonlimearity, Funkcial. Ekvac., 56 (2013), 193-224. doi: 10.1619/fesi.56.193. Google Scholar

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Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions and blow-up phenomena of mass critical fractional Schrödinger equations, Nonlinear Anal., 86 (2013), 12-29. doi: 10.1016/j.na.2013.03.002. Google Scholar

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Y. Cho, G. Hwang, S. Kwon and S. Lee, On finite time blow-up for the mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 467-479. doi: 10.1017/S030821051300142X. Google Scholar

[11]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880. doi: 10.3934/dcds.2015.35.2863. Google Scholar

[12]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions of fractional Schrödinger equations with angularly regular data, J. Differential Equations, 256 (2014), 3011-3037. doi: 10.1016/j.jde.2014.01.030. Google Scholar

[13]

Y. Cho, G. Hwang and T. Ozawa, On the focusing energy-critical fractional nonlinear Schrödinger equations, Adv. Differential Equations, 23 (2018), 161-192. Google Scholar

[14]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970. Google Scholar

[15]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399. Google Scholar

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Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. Google Scholar

[17]

M. Christ and I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C. Google Scholar

[18]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $ \mathbb R^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014. doi: 10.1002/cpa.20029. Google Scholar

[19]

V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525. Google Scholar

[20]

V. D. Dinh, On blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Commun. Pure Appl. Anal., 18 (2019), 689-708. Google Scholar

[21]

V. D. Dinh, A study on blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation, J. Math. Phys., 59 (2018), 071506, 25pp. doi: 10.1063/1.5027713. Google Scholar

[22]

B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220. doi: 10.1007/s00028-017-0397-z. Google Scholar

[23]

B. Feng, On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804. doi: 10.3934/cpaa.2018085. Google Scholar

[24]

B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Hartree equation, J. Math. Anal. Appl., 460 (2018), 352-364. doi: 10.1016/j.jmaa.2017.11.060. Google Scholar

[25]

B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499-2507. doi: 10.1016/j.camwa.2017.12.025. Google Scholar

[26]

R. L. Frank and E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in $ \mathbb R$, Acta Math..Google Scholar

[27]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. Google Scholar

[28]

J. Fröhlich, G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. doi: 10.1007/s00220-007-0272-9. Google Scholar

[29]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155. Google Scholar

[30]

Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrödinger equation in the radial case, Dyn. Partial Differ. Equ., 15 (2018), 265-282. doi: 10.4310/DPDE.2018.v15.n4.a2. Google Scholar

[31]

Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6. Google Scholar

[32]

Q. Guo and S. Zhu, Sharp threshold of blow-up and scattering for the fractional Hartree equation, J. Differential Equations, 264 (2018), 2802-2832. doi: 10.1016/j.jde.2017.11.001. Google Scholar

[33]

Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549.Google Scholar

[34]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265. Google Scholar

[35]

Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039. Google Scholar

[36]

K. Kirkpatrick, E. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. doi: 10.1007/s00220-012-1621-x. Google Scholar

[37]

N. Laskin, Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar

[38]

N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar

[39]

F. Merle and P. Raphaël, Blow up of critical norm for some radial $L^2$ super critical nonlinear Schrödinger equations, Amer. J. Math., 130 (2008), 945-978. doi: 10.1353/ajm.0.0012. Google Scholar

[40]

C. Miao, G. Xu and L. Zhao, The dynamics of the 3D radial NLS with combined terms, Comm. Math. Phys., 318 (2013), 767-808. doi: 10.1007/s00220-013-1677-2. Google Scholar

[41]

C. Miao, T. Zhao and J. Zheng, On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold, Calc. Var. Partial Differential Equations, 56 (2017), Art. 179, 39 pp. doi: 10.1007/s00526-017-1264-z. Google Scholar

[42]

C. Peng and Q. Shi, Stability of standing waves for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp. doi: 10.1063/1.5021689. Google Scholar

[43]

C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228. doi: 10.3934/dcds.2018091. Google Scholar

[44]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805. Google Scholar

[45]

H. Triebel, Theory of Function Spaces, Basel Birkhäuser, 1983. doi: 10.1007/978-3-0346-0416-1. Google Scholar

[46]

G. X. Xu and J. W. Yang, Long time dynamics of the 3D radial NLS with the combined terms, Acta Math. Sin. (Engl. Ser.), 32 (2016), 521-540. doi: 10.1007/s10114-016-5401-y. Google Scholar

[47]

J. Zhang and S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030. doi: 10.1007/s10884-015-9477-3. Google Scholar

[48]

S. Zhu, On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531. doi: 10.1016/j.jde.2016.04.007. Google Scholar

[49]

S. Zhu, Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021. doi: 10.1007/s00028-016-0363-1. Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976. Google Scholar

[2]

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

[3]

T. Boulenger, D. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603. doi: 10.1016/j.jfa.2016.08.011. Google Scholar

[4]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A. Google Scholar

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lectures Notes in Mathematics 10, New York, AMS, 2003. doi: 10.1090/cln/010. Google Scholar

[6]

X. Cheng, C. Miao and L. Zhao, Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934. doi: 10.1016/j.jde.2016.04.031. Google Scholar

[7]

Y. Cho, Short-range scattering of Hartree type fractional NLS, J. Differential Equations, 262 (2017), 116-144. doi: 10.1016/j.jde.2016.09.025. Google Scholar

[8]

Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equations with Hartree type nonlimearity, Funkcial. Ekvac., 56 (2013), 193-224. doi: 10.1619/fesi.56.193. Google Scholar

[9]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions and blow-up phenomena of mass critical fractional Schrödinger equations, Nonlinear Anal., 86 (2013), 12-29. doi: 10.1016/j.na.2013.03.002. Google Scholar

[10]

Y. Cho, G. Hwang, S. Kwon and S. Lee, On finite time blow-up for the mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 467-479. doi: 10.1017/S030821051300142X. Google Scholar

[11]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880. doi: 10.3934/dcds.2015.35.2863. Google Scholar

[12]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions of fractional Schrödinger equations with angularly regular data, J. Differential Equations, 256 (2014), 3011-3037. doi: 10.1016/j.jde.2014.01.030. Google Scholar

[13]

Y. Cho, G. Hwang and T. Ozawa, On the focusing energy-critical fractional nonlinear Schrödinger equations, Adv. Differential Equations, 23 (2018), 161-192. Google Scholar

[14]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970. Google Scholar

[15]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365. doi: 10.1142/S0219199709003399. Google Scholar

[16]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. Google Scholar

[17]

M. Christ and I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C. Google Scholar

[18]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $ \mathbb R^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014. doi: 10.1002/cpa.20029. Google Scholar

[19]

V. D. Dinh, Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525. Google Scholar

[20]

V. D. Dinh, On blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Commun. Pure Appl. Anal., 18 (2019), 689-708. Google Scholar

[21]

V. D. Dinh, A study on blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation, J. Math. Phys., 59 (2018), 071506, 25pp. doi: 10.1063/1.5027713. Google Scholar

[22]

B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220. doi: 10.1007/s00028-017-0397-z. Google Scholar

[23]

B. Feng, On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804. doi: 10.3934/cpaa.2018085. Google Scholar

[24]

B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Hartree equation, J. Math. Anal. Appl., 460 (2018), 352-364. doi: 10.1016/j.jmaa.2017.11.060. Google Scholar

[25]

B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499-2507. doi: 10.1016/j.camwa.2017.12.025. Google Scholar

[26]

R. L. Frank and E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in $ \mathbb R$, Acta Math..Google Scholar

[27]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. Google Scholar

[28]

J. Fröhlich, G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. doi: 10.1007/s00220-007-0272-9. Google Scholar

[29]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. doi: 10.1007/BF01168155. Google Scholar

[30]

Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrödinger equation in the radial case, Dyn. Partial Differ. Equ., 15 (2018), 265-282. doi: 10.4310/DPDE.2018.v15.n4.a2. Google Scholar

[31]

Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38. doi: 10.1007/s11854-014-0025-6. Google Scholar

[32]

Q. Guo and S. Zhu, Sharp threshold of blow-up and scattering for the fractional Hartree equation, J. Differential Equations, 264 (2018), 2802-2832. doi: 10.1016/j.jde.2017.11.001. Google Scholar

[33]

Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549.Google Scholar

[34]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282. doi: 10.3934/cpaa.2015.14.2265. Google Scholar

[35]

Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861. doi: 10.1016/j.jmaa.2011.09.039. Google Scholar

[36]

K. Kirkpatrick, E. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. doi: 10.1007/s00220-012-1621-x. Google Scholar

[37]

N. Laskin, Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar

[38]

N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar

[39]

F. Merle and P. Raphaël, Blow up of critical norm for some radial $L^2$ super critical nonlinear Schrödinger equations, Amer. J. Math., 130 (2008), 945-978. doi: 10.1353/ajm.0.0012. Google Scholar

[40]

C. Miao, G. Xu and L. Zhao, The dynamics of the 3D radial NLS with combined terms, Comm. Math. Phys., 318 (2013), 767-808. doi: 10.1007/s00220-013-1677-2. Google Scholar

[41]

C. Miao, T. Zhao and J. Zheng, On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold, Calc. Var. Partial Differential Equations, 56 (2017), Art. 179, 39 pp. doi: 10.1007/s00526-017-1264-z. Google Scholar

[42]

C. Peng and Q. Shi, Stability of standing waves for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp. doi: 10.1063/1.5021689. Google Scholar

[43]

C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228. doi: 10.3934/dcds.2018091. Google Scholar

[44]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805. Google Scholar

[45]

H. Triebel, Theory of Function Spaces, Basel Birkhäuser, 1983. doi: 10.1007/978-3-0346-0416-1. Google Scholar

[46]

G. X. Xu and J. W. Yang, Long time dynamics of the 3D radial NLS with the combined terms, Acta Math. Sin. (Engl. Ser.), 32 (2016), 521-540. doi: 10.1007/s10114-016-5401-y. Google Scholar

[47]

J. Zhang and S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030. doi: 10.1007/s10884-015-9477-3. Google Scholar

[48]

S. Zhu, On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531. doi: 10.1016/j.jde.2016.04.007. Google Scholar

[49]

S. Zhu, Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021. doi: 10.1007/s00028-016-0363-1. Google Scholar

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