# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022041
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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
 $$$\label{eq_abstract} (- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \text{in } \mathbb{R}^N \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(P)}$$$
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. [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. Byeon, O. 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. Cho, M. M. Fall, H. Hajaiej, P.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. Cho, H. Hajaiej, G. 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. Cingolani, M. 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. Cingolani, M. 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. Cingolani, M. 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. Coleman, V. 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. Dai, Y. 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'Acqua, T. 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'Avenia, G. 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'Avenia, G. 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 Nezza, G. 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. Felmer, A. 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. Fröhlich, B. 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. Fröhlich, T.-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. Giacomoni, D. 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. Hainzl, E. Lenzmann, M. 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. Hajaiej, P. 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] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105.  doi: 10.1002/sapm197757293. [46] E.H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684. [47] P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6. [48] H. Luo, Ground state solutions of Pohozaev type for fractional Choquard equations with general nonlinearities, Comput. Math. Appl., 77 (2019), 877-887.  doi: 10.1016/j.camwa.2018.10.024. [49] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3. [50] I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019. [51] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007. [52] V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2. [53] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1. [54] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. [55] R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600.  doi: 10.1007/BF02105068. [56] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.  doi: 10.1098/rsta.1998.0256. [57] R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. [58] Z. Shen, F. Gao and M. Yin, Ground state for nonlinear fractional Choquard equations with general nonlinearities, Math. Methods Appl. Sci., 39 (2016), 4082-4098.  doi: 10.1002/mma.3849. [59] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [60] C. A. Stuart, Existence theory for the Hartree equation, Arch. Ration. Mech. Anal., 51 (1973), 60-69.  doi: 10.1007/BF00275993. [61] K. P. Tod, The ground state energy of the Schrödinger-Newton equation, Phys. Lett. A, 280 (2001), 173-176.  doi: 10.1016/S0375-9601(01)00059-7. [62] P. Tod and I.M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002. [63] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birkäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

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. Byeon, O. 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. Cho, M. M. Fall, H. Hajaiej, P.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. Cho, H. Hajaiej, G. 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. Cingolani, M. 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. Cingolani, M. 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. Cingolani, M. 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. Coleman, V. 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. Dai, Y. 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'Acqua, T. 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'Avenia, G. 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'Avenia, G. 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 Nezza, G. 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. Felmer, A. 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. Fröhlich, B. 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. Fröhlich, T.-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. Giacomoni, D. 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. Hainzl, E. Lenzmann, M. 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. Hajaiej, P. 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] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105.  doi: 10.1002/sapm197757293. [46] E.H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.  doi: 10.1007/BF01217684. [47] P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6. [48] H. Luo, Ground state solutions of Pohozaev type for fractional Choquard equations with general nonlinearities, Comput. Math. Appl., 77 (2019), 877-887.  doi: 10.1016/j.camwa.2018.10.024. [49] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3. [50] I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019. [51] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007. [52] V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2. [53] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1. [54] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. [55] R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600.  doi: 10.1007/BF02105068. [56] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.  doi: 10.1098/rsta.1998.0256. [57] R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. [58] Z. Shen, F. Gao and M. Yin, Ground state for nonlinear fractional Choquard equations with general nonlinearities, Math. Methods Appl. Sci., 39 (2016), 4082-4098.  doi: 10.1002/mma.3849. [59] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [60] C. A. Stuart, Existence theory for the Hartree equation, Arch. Ration. Mech. Anal., 51 (1973), 60-69.  doi: 10.1007/BF00275993. [61] K. P. Tod, The ground state energy of the Schrödinger-Newton equation, Phys. Lett. A, 280 (2001), 173-176.  doi: 10.1016/S0375-9601(01)00059-7. [62] P. Tod and I.M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002. [63] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birkäuser, 1996. doi: 10.1007/978-1-4612-4146-1.
 [1] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [2] Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235 [3] Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136 [4] Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131 [5] Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499 [6] Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091 [7] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445 [8] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [9] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 [10] Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 [11] Peng Chen, Xianhua Tang. Ground states for a system of nonlinear Schrödinger equations with singular potentials. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 5105-5136. doi: 10.3934/dcds.2022088 [12] Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure and Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008 [13] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [14] Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121 [15] Xavier Cabré. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 331-359. doi: 10.3934/dcds.2002.8.331 [16] Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3085-3096. doi: 10.3934/dcdss.2021079 [17] Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure and Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015 [18] Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $p$-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026 [19] Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068 [20] Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991

2021 Impact Factor: 1.865