1Bahouri, H., Chemin, J. Y. and Danchin, R., Fourier analysis and nonlinear partial differential equations, A Series of Comprehensive Studies in Mathematics, vol. 343, (Berlin: Springer, 2011).

2Bensouilah, A., Dinh, V. D. and Zhu, S.. On stability and instability of standing waves for the nonlinear Schrödinger equation with an inverse-square potential. J. Math. Phys. 104 (2018), 101505.

3Cabré, X. and Sire, Y.. Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53.

4Cabré, X. and Sire, Y.. Nonlinear equations for fractional Laplacians, II: existence, uniqueness, and qualitative properties of solutions. Trans. Amer. Math. Soc. 367 (2015), 911–941.

5Caffarelli, L. and Silvestre, L.. An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32 (2007), 1245–1260.

6Caffarelli, L., Salsa, S. and Silvestre, L.. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171 (2008), 425–461.

7Cafferalli, L., Roquejoffre, J. M. and Sire, Y.. Variational problems with free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12 (2010), 1151–1179.

8Cazenave, T., Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10 (Providence, RI: American Mathematical Society, 2003).

9Cho, Y. and Ozawa, T.. Sobolev inequalities with symmetry. Commun. Contemp. Math. 11 (2009), 355–365.

10Di Nezza, E., Palatucci, G. and Valdinoci, E.. Hitchhiker's guide to the fractional Sobolev spaces. Bull. des Sci. Math. 136 (2012), 521–573.

11Du, M., Tian, L., Wang, J. and Zhang, F.. Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials. Proc. Roy. Soc. Edinburgh Sect. A 149 (2018), 617–653.

12Felmer, P., Quaas, A. and Tan, J.. Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237–1262.

13Feng, B.. Ground states for the fractional Schrödinger equation. Electron. J. Differ. Equ. 127 (2013), 1–11.

14Frank, R. and Lenzmann, E.. Uniqueness and nondegeneracy of ground states for (−Δ)^{s} + *Q* − *Q* ^{α+1} = 0 in ℝ. Acta. Math. 210 (2013), 261–318.

15Frank, R., Lenzmann, E. and Silvestre, L.. Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math. 69 (2016), 1671–1726.

16Fröhlich, J., Jonsson, G. and Lenzmann, E.. Boson stars as solitary waves. Commun. Math. Phys. 274 (2007), 1–30.

17Guo, Y. and Seiringer, R.. On the mass concentration for Bose-Einstein condensates with attractive interactions. Lett. Math. Phys. 104 (2014), 141–156.

18Guo, H. and Zhou, H. S.. A constrained variational problem arising in attractive Bose-Einstein condensate with ellipse-shaped potential. Appl. Math. Lett. 87 (2019), 35–41.

19Guo, Y., Zeng, X. and Zhou, H. S.. Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with righ-shaped potentials. Ann. Inst. Henri Poincaré Non Lineaire Anal. 33 (2016), 809–828.

20Guo, Y., Wang, Z. Q., Zeng, X. and Zhou, H. S.. Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials. Nonlinearity 31 (2018), 957.

21Guo, Y., Luo, Y. and Yang, W., Refined mass concentration of rotating Bose-Einstein condensates with attractive interactions, Preprint 2019. arXiv:1901.09619.

22He, Q. and Long, W.. The concentration of solutions to a fractional Schrödinger equation. Z. Angew. Math. Phys. 67: 9 (2016).

23Kirkpatrick, K., Lenzmann, E. and Staffilani, G.. On the continuum limit for discrete NLS with long-range lattice interactions. Commun. Math. Phys. 317 (2013), 563–591.

24Laskin, N.. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268 (2000), 298–304.

25Laskin, N.. Fractional Schrödinger equations. Phys. Rev. E 66 (2002), 056108.

26Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case: Part 1. Ann. Inst. Henri Poincaré 1 (1984), 109–145.

27Lions, P. L.. The concentration-compactness principle in the calculus of variations. The locally compact case: Part 2. Ann. Inst. Henri Poincaré 1 (1984), 223–283.

28Ionescu, A. D. and Pusateri, F.. Nonlinear fractional Schrödinger equations in one dimensions. J. Funct. Anal. 266 (2014), 139–176.

29Maeda, M.. On the symmetry of the ground states of nonlinear Schrödinger equation with potential. Adv. Nonlinear Stud. 10 (2010), 895–925.

30Park, Y. J.. Fractional Polya-Szegö inequality. J. Chungcheong Math. Soc. 24 (2011), 267–271.

31Phan, T. V.. Blow-up profile of Bose-Einstein condensate with singular potentials. J. Math. Phys. 58 (2017), 072301.

32Sire, Y. and Valdinoci, E.. Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result. J. Funct. Anal. 256 (2009), 1842–1864.

33Wang, Q. and Zhao, D.. Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials. J. Differ. Equ. 262 (2017), 2684–2704.

34Zhang, J.. Stability of attractive Bose-Einstein condensates. J. Stat. Phys. 101 (2000), 731–746.