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On fractional Schrödinger equations with Hartree type nonlinearities

  • Received: 10 August 2021 Accepted: 17 November 2021 Published: 30 December 2021
  • MSC : 35B38, 35B40, 35J20, 35Q40, 35Q55, 35R09, 35R11, 45M05

  • Goal of this paper is to study the following doubly nonlocal equation

    $(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad {\rm{in}}\;{\mathbb{R}^N}\qquad\qquad\qquad\qquad ({\rm{P}}) $

    in the case of general nonlinearities $ F \in C^1(\mathbb{R}) $ of Berestycki-Lions type, when $ N \geq 2 $ and $ \mu > 0 $ is fixed. Here $ (-\Delta)^s $, $ s \in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{\alpha} $, $ \alpha \in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [23,61].

    Citation: Silvia Cingolani, Marco Gallo, Kazunaga Tanaka. On fractional Schrödinger equations with Hartree type nonlinearities[J]. Mathematics in Engineering, 2022, 4(6): 1-33. doi: 10.3934/mine.2022056

    Related Papers:

  • Goal of this paper is to study the following doubly nonlocal equation

    $(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad {\rm{in}}\;{\mathbb{R}^N}\qquad\qquad\qquad\qquad ({\rm{P}}) $

    in the case of general nonlinearities $ F \in C^1(\mathbb{R}) $ of Berestycki-Lions type, when $ N \geq 2 $ and $ \mu > 0 $ is fixed. Here $ (-\Delta)^s $, $ s \in (0, 1) $, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $ I_{\alpha} $, $ \alpha \in (0, N) $. We prove existence of ground states of (P). Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in [23,61].



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    [1] C. Argaez, M. Melgaard, Solutions to quasi-relativistic multi-configurative Hartree–Fock equations in quantum chemistry, Nonlinear Anal. Theor., 75 (2012), 384–404. doi: 10.1016/j.na.2011.08.038
    [2] D. R. Adams, L. I. Hedberg, Function spaces and potential theory, Berlin Heidelberg: Springer, 1996.
    [3] M. Badiale, L. Pisani, S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, Nonlinear Differ. Equ. Appl., 18 (2011), 369–405. doi: 10.1007/s00030-011-0100-y
    [4] P. Belchior, H. Bueno, O. H. Miyagaki, G. A. Pereira, Remarks about a fractional Choquard equation: ground state, regularity and polynomial decay, Nonlinear Anal., 164 (2017), 38–53. doi: 10.1016/j.na.2017.08.005
    [5] H. Berestycki, 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] C. Bucur, E. Valdinoci, Nonlocal diffusion and applications, Springer Nature Switzerland AG, 2016.
    [7] J. Byeon, O. Kwon, J. Seok, Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity, 30 (2017), 1659–1681. doi: 10.1088/1361-6544/aa60b4
    [8] H. Brezis, T. Kato, Remarks on the Schrödinger operator with singular complex potential, J. Math. Pure. Appl., 58 (1979), 137–151.
    [9] X. Cabré, 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
    [10] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq., 32 (2007), 1245–1260. doi: 10.1080/03605300600987306
    [11] X. Chang, Z.-Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearities, Nonlinearity, 26 (2013), 479–494. doi: 10.1088/0951-7715/26/2/479
    [12] Y. Cho, G. Hwang, H. Hajaiej, T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkc. Ekvacioj, 56 (2013), 193–224. doi: 10.1619/fesi.56.193
    [13] Y. Cho, M. M. Fall, H. Hajaiej, P.$ $A. Markowich, S. Trabelsi, Orbital stability of standing waves of a class of fractional Schrödinger equations with Hartree-type nonlinearity, Anal. Appl., 15 (2017), 699–729. doi: 10.1142/S0219530516500056
    [14] S. Cingolani, M. Clapp, 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
    [15] S. Cingolani, M. Gallo, On the fractional NLS equation and the effects of the potential well's topology, Adv. Nonlinear Stud., 21 (2021), 1–40. doi: 10.1515/ans-2020-2114
    [16] S. Cingolani, M. Gallo, K. Tanaka, Normalized solutions for fractional nonlinear scalar field equation via Lagrangian formulation, Nonlinearity, 34 (2021), 4017–4056. doi: 10.1088/1361-6544/ac0166
    [17] S. Cingolani, M. Gallo, K. Tanaka, Symmetric ground states for doubly nonlocal equations with mass constraint, Symmetry, 13 (2021), 1199. doi: 10.3390/sym13071199
    [18] S. Cingolani, K. Tanaka, Deformation argument under PSP condition and applications, Anal. Theory Appl., 37 (2021), 191–208. doi: 10.4208/ata.2021.pr80.03
    [19] S. Cingolani, M. Gallo, K. Tanaka, Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities, Calc. Var., in press.
    [20] M. Clapp, 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
    [21] S. Coleman, V. Glaser, 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
    [22] A. Dall'Acqua, T. Ø. Sørensen, 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, M. Squassina, On the fractional Choquard equations, Math. Mod. Meth. Appl. Sci., 25 (2015), 1447–1476. doi: 10.1142/S0218202515500384
    [24] P. D'Avenia, G. Siciliano, 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] E. Di Nezza, G. Palatucci, 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
    [26] S. Dipierro, M. Medina, E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $ \mathbb{R}^n$, Pisa: Edizioni della Normale, 2017.
    [27] L. Dong, D. Liu, W. Qi, L. Wang, H. Zhou, P. Peng, et al., Necklace beams carrying fractional angular momentum in fractional systems with a saturable nonlinearity, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), 105840. doi: 10.1016/j.cnsns.2021.105840
    [28] P. Felmer, A. Quaas, J. Tan, Positive solutions of the nonlinear Schrödinger equations with the fractional Laplacian, P. Roy. Soc. Edinb. A, 142 (2012), 1237–1262. doi: 10.1017/S0308210511000746
    [29] R. L. Frank, E. Lenzmann, On ground states for the $L^2$-critical boson star equation, arXiv: 0910.2721.
    [30] R. L. Frank, E. Lenzmann, L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69 (2016), 1671–1726. doi: 10.1002/cpa.21591
    [31] J. Fröhlich, B. L. G. Jonsson, E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys., 274 (2007), 1–30. doi: 10.1007/s00220-007-0272-9
    [32] J. Fröhlich, E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, In: Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2003-2004), talk no. 18, 26.
    [33] J. Fröhlich, T.-P. Tsai, H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Commun. Math. Phys., 225 (2002), 223–274. doi: 10.1007/s002200100579
    [34] M. Gallo, Multiplicity and concentration results for local and fractional NLS equations with critical growth, Adv. Differential Equ., 26 (2021), 397–424.
    [35] J. Giacomoni, D. Goel, K. Sreenadh, Regularity results on a class of doubly nonlocal problems, J. Differ. Equations, 268 (2020), 5301–5328. doi: 10.1016/j.jde.2019.11.009
    [36] Q. Guo, S. Zhu, Sharp threshold of blow-up and scattering for the fractional Hartree equation, J. Differ. Equations, 264 (2018), 2802–2832. doi: 10.1016/j.jde.2017.11.001
    [37] C. Hainzl, E. Lenzmann, M. Lewin, 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
    [38] H. Hajaiej, P. A. Markowich, S. Trabelsi, Multiconfiguration Hartree-Fock theory for pseudorelativistic systems: the time-dependent case, Math. Mod. Meth. Appl. Sci., 24 (2014), 599–626. doi: 10.1142/S0218202513500619
    [39] S. Herr, E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal. Theor., 97 (2014), 125–137. doi: 10.1016/j.na.2013.11.023
    [40] J. Hirata, 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
    [41] N. Ikoma, Existence of solutions of scalar field equations with fractional operator, J. Fixed Point Theory Appl., 19 (2017), 649–690. doi: 10.1007/s11784-016-0369-x
    [42] N. Ikoma, Erratum to: Existence of solutions of scalar field equations with fractional operator, J. Fixed Point Theory Appl., 19 (2017), 1649–1652. doi: 10.1007/s11784-017-0427-z
    [43] N. Ikoma, K. Tanaka, A note on deformation argument for $L^2$ constraint problems, Adv. Differential Equ., 24 (2019), 609–646.
    [44] L. Jeanjean, K. Tanaka, A remark on least energy solutions in $ \mathbb{R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399–2408.
    [45] K. Kirkpatrick, E. Lenzmann, G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys., 317 (2013), 563–591. doi: 10.1007/s00220-012-1621-x
    [46] C. Klein, C. Sparber, P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. Royal Soc. A, 470 (2014), 20140364. doi: 10.1098/rspa.2014.0364
    [47] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Rev. A, 268 (2000), 56–108.
    [48] E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 2 (2007), 43–64.
    [49] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1–27. doi: 10.2140/apde.2009.2.1
    [50] E. Lenzmann, M. Lewin, On singularity formation for the $L^2$-critical Boson star equation, Nonlinearity, 24 (2011), 3515–3540. doi: 10.1088/0951-7715/24/6/008
    [51] 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
    [52] E. H. Lieb, M. Loss, Analysis, USA: American Mathematical Society, 2001.
    [53] E. H. Lieb, H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147–174. doi: 10.1007/BF01217684
    [54] 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
    [55] S. Longhi, Fractional Schrödinger equation in optics, Optics Lett., 40 (2015), 1117–1120. doi: 10.1364/OL.40.001117
    [56] J. Lu, V. Moroz, C. B. Muratov, Orbital-free density functional theory of out-of-plane charge screening in graphene, J. Nonlinear. Sci., 25 (2015), 1391–1430. doi: 10.1007/s00332-015-9259-4
    [57] 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
    [58] L. Ma, 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
    [59] I. M. Moroz, R. Penrose, P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Class. Quantum Grav., 15 (1998), 2733–2742. doi: 10.1088/0264-9381/15/9/019
    [60] V. Moroz, 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
    [61] V. Moroz, J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557–6579.
    [62] V. Moroz, 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
    [63] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Berlin: Akademie Verlag, 1954.
    [64] R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581–600. doi: 10.1007/BF02105068
    [65] R. Penrose, Quantum computation, entanglement and state reduction, Phil. Trans. R. Soc. A, 356 (1998), 1927–1939. doi: 10.1098/rsta.1998.0256
    [66] R. Penrose, The road to reality. A complete guide to the laws of the universe, New York: Alfred A. Knopf Inc., 2005.
    [67] Z. Shen, F. Gao, M. Yin, Ground state for nonlinear fractional Choquard equations with general nonlinearities, Math. Method. Appl. Sci., 39 (2016), 4082–4098. doi: 10.1002/mma.3849
    [68] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2006), 67–112.
    [69] C. Stuart, Existence theory for the {H}artree equation, Arch. Rational Mech. Anal., 51 (1973), 60–69. doi: 10.1007/BF00275993
    [70] 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
    [71] P. Tod, I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201–216. doi: 10.1088/0951-7715/12/2/002
    [72] Z. Yang, F. Zhao, Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth, Adv. Nonlinear Anal., 10 (2021), 732–774.
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