doi: 10.3934/dcdss.2022041
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On some qualitative aspects for doubly nonlocal equations

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

*Corresponding author: Silvia Cingolani

Dedicated to the memory of Professor Rosella Mininni, a brilliant mathematician, and a very nice person

Received  October 2021 Early access February 2022

In this paper we investigate some qualitative properties of the solutions to the following doubly nonlocal equation
$ \begin{equation} \label{eq_abstract} (- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \text{in } \mathbb{R}^N \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(P)}\end{equation} $
where
$ N \geq 2 $
,
$ s\in (0, 1) $
,
$ \alpha \in (0, N) $
,
$ \mu>0 $
is fixed,
$ (-\Delta)^s $
denotes the fractional Laplacian and
$ I_{\alpha} $
is the Riesz potential. Here
$ F \in C^1(\mathbb{R}) $
stands for a general nonlinearity of Berestycki-Lions type. We obtain first some regularity result for the solutions of (P). Then, by assuming
$ F $
odd or even and positive on the half-line, we get constant sign and radial symmetry of the Pohozaev ground state solutions related to equation (P). In particular, we extend some results contained in [23]. Similar qualitative properties of the ground states are obtained in the limiting case
$ s = 1 $
, generalizing some results by Moroz and Van Schaftingen in [52] when
$ F $
is odd.
Citation: Silvia Cingolani, Marco Gallo. On some qualitative aspects for doubly nonlocal equations. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022041
References:
[1]

C. Argaez and M. Melgaard, Solutions to quasi-relativistic multi-configurative Hartree-Fock equations in quantum chemistry, Nonlinear Anal., 75 (2012), 384-404.  doi: 10.1016/j.na.2011.08.038.

[2]

T. Bartsch, Y. Liu and Z. Liu, Normalized solutions for a class of nonlinear Choquard equations, SN Partial Differ. Equ. Appl., 1 (2020), Paper No. 34, 25 pp. doi: 10.1007/s42985-020-00036-w.

[3]

W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Natl. Acad. Sci. USA, 89 (1992), 4816-4819.  doi: 10.1073/pnas.89.11.4816.

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J. Benedikt, V. Bobkov, R. N. Dhara and P. Girg, Nonradiality of second eigenfunctions of the fractional Laplacian in a ball, arXiv: 2102.08298, (2021), pp. 13.

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H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[6]

J. ByeonO. Kwon and J. Seok, Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity, 30 (2017), 1659-1681.  doi: 10.1088/1361-6544/aa60b4.

[7]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

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Y.-H. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.  doi: 10.1088/0951-7715/29/6/1827.

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Y. ChoM. M. FallH. HajaiejP.A. Markowich and S. Trabelsi, Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity, Anal. Appl. (Singap.), 15 (2017), 699-729.  doi: 10.1142/S0219530516500056.

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Y. ChoH. HajaiejG. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcial. Ekvac., 56 (2013), 193-224.  doi: 10.1619/fesi.56.193.

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S. CingolaniM. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.  doi: 10.1007/s00033-011-0166-8.

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S. CingolaniM. Gallo and K. Tanaka, Normalized solutions for fractional nonlinear scalar field equation via Lagrangian formulation, Nonlinearity, 34 (2021), 4017-4056.  doi: 10.1088/1361-6544/ac0166.

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S. Cingolani, M. Gallo and K. Tanaka, Symmetric ground states for doubly nonlocal equations with mass constraint, Symmetry, 13 (2021), article ID 1199, 1–17. doi: 10.3390/sym13071199.

[14]

S. Cingolani, M. Gallo and K. Tanaka, Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities, Calc. Var. Partial Differential Equations, 61 (2022), Paper No. 68. doi: 10.1007/s00526-021-02182-4.

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S. CingolaniM. Gallo and K. Tanaka, On fractional Schrödinger equations with Hartree type nonlinearities, Mathematics in Engineering, 4 (2022), 1-33.  doi: 10.3934/mine.2022056.

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S. Cingolani and K. Tanaka, Ground state solutions for the nonlinear Choquard equation with prescribed mass, in Geometric Properties for Parabolic and Elliptic PDE's, Springer INdAM Series, 47 (2021), 23–41. doi: 10.1007/978-3-030-73363-6_2.

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S. Cingolani and K. Tanaka, Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well, Rev. Mat. Iberoam., 35 (2019), 1885-1924.  doi: 10.4171/rmi/1105.

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M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15.  doi: 10.1016/j.jmaa.2013.04.081.

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R. Clemente, J.C. de Albuquerque and E. Barboza, Existence of solutions for a fractional Choquard-type equation in $\mathbb{R}$ with critical exponential growth, Z. Angew. Math. Phys., 72 (2021), Paper No. 16, 13 pp. doi: 10.1007/s00033-020-01447-w.

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W. DaiY. Fang and G. Qin, Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes, J. Differential Equations, 265 (2018), 2044-2063.  doi: 10.1016/j.jde.2018.04.026.

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A. Dall'AcquaT. Østergaard Sørensen and E. Stockmeyer, Hartree-Fock theory for pseudorelativistic atoms, Ann. Henri Poincaré, 9 (2008), 711-742.  doi: 10.1007/s00023-008-0370-z.

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P. d'AveniaG. Siciliano and M. Squassina, On the fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.

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P. d'AveniaG. Siciliano and M. Squassina, Existence results for a doubly nonlocal equation, São Paulo J. Math. Sci., 9 (2015), 311-324.  doi: 10.1007/s40863-015-0023-3.

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F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Universitext, Springer-Verlag, London, 2012. doi: 10.1007/978-1-4471-2807-6.

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E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

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L. Dong, D. Liu, W. Qi, L. Wang, H. Zhou, P. Peng and C. Huang, Necklace beams carrying fractional angular momentum in fractional systems with a saturable nonlinearity, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), article ID 105840, 8 pp. doi: 10.1016/j.cnsns.2021.105840.

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N. du Plessis, Some theorems about the Riesz fractional integral, Trans. Am. Math. Soc., 80 (1955), 124-134.  doi: 10.1090/S0002-9947-1955-0086938-3.

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

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P. FelmerA. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equations with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

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

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J. Fröhlich and E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, in Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau (2004), talk no. 18, 26 pp.

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J. FröhlichT.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys., 225 (2002), 223-274.  doi: 10.1007/s002200100579.

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J. GiacomoniD. Goel and K. Sreenadh, Regularity results on a class of doubly nonlocal problems, J. Differential Equations, 268 (2020), 5301-5328.  doi: 10.1016/j.jde.2019.11.009.

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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.

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H. HajaiejP. A. Markowich and S. Trabelsi, Multiconfiguration Hartree-Fock Theory for pseudorelativistic systems: The time-dependent case, Math. Models Methods Appl. Sci., 24 (2014), 599-626.  doi: 10.1142/S0218202513500619.

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S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137.  doi: 10.1016/j.na.2013.11.023.

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J. Hirata and K. Tanaka, Nonlinear scalar field equations with $L^2$ constraint: Mountain pass and symmetric mountain pass approaches, Adv. Nonlinear Stud., 19 (2019), 263-290.  doi: 10.1515/ans-2018-2039.

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N. Ikoma and K. Tanaka, A note on deformation argument for $L^2$ constraint problems, Adv. Differential Equations, 24 (2019), 609-646. 

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E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.

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

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

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show all references

References:
[1]

C. Argaez and M. Melgaard, Solutions to quasi-relativistic multi-configurative Hartree-Fock equations in quantum chemistry, Nonlinear Anal., 75 (2012), 384-404.  doi: 10.1016/j.na.2011.08.038.

[2]

T. Bartsch, Y. Liu and Z. Liu, Normalized solutions for a class of nonlinear Choquard equations, SN Partial Differ. Equ. Appl., 1 (2020), Paper No. 34, 25 pp. doi: 10.1007/s42985-020-00036-w.

[3]

W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Natl. Acad. Sci. USA, 89 (1992), 4816-4819.  doi: 10.1073/pnas.89.11.4816.

[4]

J. Benedikt, V. Bobkov, R. N. Dhara and P. Girg, Nonradiality of second eigenfunctions of the fractional Laplacian in a ball, arXiv: 2102.08298, (2021), pp. 13.

[5]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[6]

J. ByeonO. Kwon and J. Seok, Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity, 30 (2017), 1659-1681.  doi: 10.1088/1361-6544/aa60b4.

[7]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

[8]

Y.-H. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.  doi: 10.1088/0951-7715/29/6/1827.

[9]

Y. ChoM. M. FallH. HajaiejP.A. Markowich and S. Trabelsi, Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity, Anal. Appl. (Singap.), 15 (2017), 699-729.  doi: 10.1142/S0219530516500056.

[10]

Y. ChoH. HajaiejG. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcial. Ekvac., 56 (2013), 193-224.  doi: 10.1619/fesi.56.193.

[11]

S. CingolaniM. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.  doi: 10.1007/s00033-011-0166-8.

[12]

S. CingolaniM. Gallo and K. Tanaka, Normalized solutions for fractional nonlinear scalar field equation via Lagrangian formulation, Nonlinearity, 34 (2021), 4017-4056.  doi: 10.1088/1361-6544/ac0166.

[13]

S. Cingolani, M. Gallo and K. Tanaka, Symmetric ground states for doubly nonlocal equations with mass constraint, Symmetry, 13 (2021), article ID 1199, 1–17. doi: 10.3390/sym13071199.

[14]

S. Cingolani, M. Gallo and K. Tanaka, Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities, Calc. Var. Partial Differential Equations, 61 (2022), Paper No. 68. doi: 10.1007/s00526-021-02182-4.

[15]

S. CingolaniM. Gallo and K. Tanaka, On fractional Schrödinger equations with Hartree type nonlinearities, Mathematics in Engineering, 4 (2022), 1-33.  doi: 10.3934/mine.2022056.

[16]

S. Cingolani and K. Tanaka, Ground state solutions for the nonlinear Choquard equation with prescribed mass, in Geometric Properties for Parabolic and Elliptic PDE's, Springer INdAM Series, 47 (2021), 23–41. doi: 10.1007/978-3-030-73363-6_2.

[17]

S. Cingolani and K. Tanaka, Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well, Rev. Mat. Iberoam., 35 (2019), 1885-1924.  doi: 10.4171/rmi/1105.

[18]

M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15.  doi: 10.1016/j.jmaa.2013.04.081.

[19]

R. Clemente, J.C. de Albuquerque and E. Barboza, Existence of solutions for a fractional Choquard-type equation in $\mathbb{R}$ with critical exponential growth, Z. Angew. Math. Phys., 72 (2021), Paper No. 16, 13 pp. doi: 10.1007/s00033-020-01447-w.

[20]

S. ColemanV. Glaser and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Commun. Math. Phys., 58 (1978), 211-221.  doi: 10.1007/BF01609421.

[21]

W. DaiY. Fang and G. Qin, Classification of positive solutions to fractional order Hartree equations via a direct method of moving planes, J. Differential Equations, 265 (2018), 2044-2063.  doi: 10.1016/j.jde.2018.04.026.

[22]

A. Dall'AcquaT. Østergaard Sørensen and E. Stockmeyer, Hartree-Fock theory for pseudorelativistic atoms, Ann. Henri Poincaré, 9 (2008), 711-742.  doi: 10.1007/s00023-008-0370-z.

[23]

P. d'AveniaG. Siciliano and M. Squassina, On the fractional Choquard equations, Math. Models Methods Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.

[24]

P. d'AveniaG. Siciliano and M. Squassina, Existence results for a doubly nonlocal equation, São Paulo J. Math. Sci., 9 (2015), 311-324.  doi: 10.1007/s40863-015-0023-3.

[25]

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Universitext, Springer-Verlag, London, 2012. doi: 10.1007/978-1-4471-2807-6.

[26]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[27]

L. Dong, D. Liu, W. Qi, L. Wang, H. Zhou, P. Peng and C. Huang, Necklace beams carrying fractional angular momentum in fractional systems with a saturable nonlinearity, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), article ID 105840, 8 pp. doi: 10.1016/j.cnsns.2021.105840.

[28]

N. du Plessis, Some theorems about the Riesz fractional integral, Trans. Am. Math. Soc., 80 (1955), 124-134.  doi: 10.1090/S0002-9947-1955-0086938-3.

[29]

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

[30]

P. FelmerA. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equations with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[31]

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

[32]

J. Fröhlich and E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, in Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau (2004), talk no. 18, 26 pp.

[33]

J. FröhlichT.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys., 225 (2002), 223-274.  doi: 10.1007/s002200100579.

[34]

J. GiacomoniD. Goel and K. Sreenadh, Regularity results on a class of doubly nonlocal problems, J. Differential Equations, 268 (2020), 5301-5328.  doi: 10.1016/j.jde.2019.11.009.

[35]

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.

[36]

C. HainzlE. LenzmannM. Lewin and B. Schlein, On blowup for time-dependent generalized Hartree-Fock equations, Ann. Henri Poincaré, 11 (2010), 1023-1052.  doi: 10.1007/s00023-010-0054-3.

[37]

H. HajaiejP. A. Markowich and S. Trabelsi, Multiconfiguration Hartree-Fock Theory for pseudorelativistic systems: The time-dependent case, Math. Models Methods Appl. Sci., 24 (2014), 599-626.  doi: 10.1142/S0218202513500619.

[38]

S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137.  doi: 10.1016/j.na.2013.11.023.

[39]

J. Hirata and K. Tanaka, Nonlinear scalar field equations with $L^2$ constraint: Mountain pass and symmetric mountain pass approaches, Adv. Nonlinear Stud., 19 (2019), 263-290.  doi: 10.1515/ans-2018-2039.

[40]

N. Ikoma and K. Tanaka, A note on deformation argument for $L^2$ constraint problems, Adv. Differential Equations, 24 (2019), 609-646. 

[41]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Rev. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[42]

E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.

[43]

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

[44]

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

[45]

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