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Normalized Ground States for the Critical Fractional Choquard Equation with a Local Perturbation

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Abstract

In this paper, we study the critical fractional Choquard equation with a local perturbation \((-\Delta )^su=\lambda u+\mu |u|^{q-2}u+(I_{\alpha }*|u|^{2^*_{\alpha ,s}})|u|^{2^*_{\alpha ,s}-2}u,~~x\in {\mathbb {R}}^N,\) having prescribed mass \(\int _{{\mathbb {R}}^N}u^2\text {d}x=a^2,\) where \(I_{\alpha }(x)\) is the Riesz potential, \(s\in (0,1), N>2s, 0<\alpha<\min \{N,4s\}, 2<q<2^*_s=\frac{2N^{}}{N-2s}\) is the fractional critical Sobolev exponent, and \(2^*_{\alpha ,s}=\frac{2N-\alpha }{N-2s}\) is the fractional Hardy–Littlewood–Sobolev critical exponent, \(a>0,\ \mu \in {\mathbb {R}}.\). Under some \(L^2\)-subcritical, \(L^2\)-critical and \(L^2\)-supercritical perturbation \(\mu |u|^{q-2}u\), respectively, we prove several existence and non-existence results. The qualitative behavior of the ground states as \(\mu \rightarrow 0^+\) is also studied. The mathematical analysis carried out in this paper can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions for fractional Choquard equation. In this framework, several related results are extended and improved.

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References

  1. Ackermann, N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Applebaum, D.: Lévy processes: from probability to finance and quantum groups. Notices Am. Math. Soc. 51, 1336–1347 (2004)

    MATH  Google Scholar 

  3. Appolloni, L., Secchi, S.: Normalized solutions for the fractional NLS with mass supercritical nonlinearity. J. Differ. Equ. 286, 248–283 (2021)

    MathSciNet  MATH  Google Scholar 

  4. Argaez, C., Melgaard, M.: Solutions to quasi-relativistic multi-configurative Hartree-Fock equations in quantum chemistry. Nonlinear Anal. TMA 75, 384–404 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Bartsch, T., Liu, Y., Liu, Z.: Normalized solutions for a class of nonlinear Choquard equations. SN Partial Differ. Equ. Appl. 34, 24 (2020)

    MATH  Google Scholar 

  6. Bartsch, T., Zhong, X., Zou, W.: Normalized solutions for a coupled Schrödinger system. Math. Ann. 380, 1713–1740 (2021)

    MathSciNet  MATH  Google Scholar 

  7. Bellazzini, J., Ghimenti, M., Mercuri, C., Moroz, V., van Schaftingen, J.: Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces. Trans. Am. Math. Soc. 370, 8285–8310 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–345 (1983)

    MathSciNet  MATH  Google Scholar 

  9. Byeon, J., Kwon, O., Seok, J.: Nonlinear scalar field equations involving the fractional Laplacian. Nonlinearity 30, 1659–1681 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Cao, D., Jia, H., Luo, X.: Standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials. J. Differ. Equ. 276, 228–263 (2021)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Cassani, D., Zhang, J.: Choquard-type equations with Hardy-Littlewood-Sobolev upper-critical growth. Adv. Nonlinear Anal. 8(1), 1184–1212 (2019)

    MathSciNet  MATH  Google Scholar 

  13. Chang, X., Wang, Z.-Q.: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearities. Nonlinearity 26, 479–494 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Cingolani, S., Gallo, M., Tanaka, K.: On fractional Schrödinger equations with Hartree type nonlinearities. http://arxiv.org/abs/2110.0753vl, [math.AP]. 14 Oct. (2021)

  15. Cingolani, S., Gallo, M., Tanaka, K.: Symmetric ground states for doubly nonlocal equations with mass constraint. Symmetry 13, 1–17 (2021)

    Google Scholar 

  16. Cingolani, S., Gallo, M., Tanaka, K.: Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation. Nonlinearity 34, 4017–4056 (2021)

    MathSciNet  MATH  Google Scholar 

  17. Cotsiolis, A., Tavoularis, N.K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295, 225–236 (2004)

    MathSciNet  MATH  Google Scholar 

  18. d’Avenia, P., Siciliano, G., Squassina, M.: On fractional Choquard equations. Math. Models Methods Appl. Sci. 25, 1447–476 (2015)

    MathSciNet  MATH  Google Scholar 

  19. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equations with the fractional Laplacian. Proc. Roy. Soc. Edinb. Sect. A 142, 1237–1262 (2012)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Frölich, J., Lenzmann, E.: Mean-field limit of quantum Bose gases and nonlinear Hartree equation. In: Sémin. Équations aux Dérivées Partielles, École Polytechnique, Palaiseau, talk no. 18, p. 26 (2004)

  23. Frölich, J., Tsai, T.-P., Yau, H.-T.: On the point-particle (Newtonian) limit of the non-linear Hartree equation. Commun. Math. Phys. 225, 223–274 (2002)

    MathSciNet  Google Scholar 

  24. Frölich, J., Jonsson, B.L.G., Lenzmann, E.: Boson stars as solitary waves. Commun. Math. Phys. 274, 1–30 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Ghoussoub, N.: Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics, vol. 107. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  26. Guo, Q., Zhu, S.: Sharp threshold of blow-up and scattering for the fractional Hartree equation. J. Differ. Equ. 264, 2802–2832 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Hajaiej, H., Markowich, P.A., Trabelsi, S.: Multiconfiguration Hartree-Fock Theory for pseudorelativistic systems: the time-dependent case. Math. Models Methods Appl. Sci. 24, 599–626 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Hainzl, C., Lenzmann, E., Lewin, M., Schlein, B.: On blowup for time-dependent generalized Hartree-Fock equations. Ann. Henri Poincaré 11, 1023–1052 (2010)

    MathSciNet  MATH  Google Scholar 

  29. He, X., Rădulescu, V.D.: Small linear perturbations of fractional Choquard equations with critical exponent. J. Differ. Equ. 282, 481–540 (2021)

    MathSciNet  MATH  Google Scholar 

  30. Herr, S., Lenzmann, E.: The Boson star equation with initial data of low regularity. Nonlinear Anal. TMA 97, 125–137 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliptic equation. Nonlinear Anal. TMA 28, 1633–1659 (1997)

    MathSciNet  MATH  Google Scholar 

  32. Jeanjean, L., Lu, S.S.: A mass supercritical problem revisited. Calc. Var. Partial Differ. Equ. 59(5), 174, 42 (2020)

    MathSciNet  MATH  Google Scholar 

  33. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Rev. A 268, 56–108 (2000)

    MathSciNet  MATH  Google Scholar 

  34. Lenzmann, E.: Well-posedness for semi-relativistic Hartree equations of critical type. Math. Phys. Anal. Geom. 2, 43–64 (2007)

    MathSciNet  MATH  Google Scholar 

  35. Lenzmann, E.: Uniqueness of ground states for pseudorelativistic Hartree equations. Anal. PDE 2, 1–27 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Lenzmann, E., Lewin, M.: On singularity formation for the \(L^2\)-critical Boson star equation. Nonlinearity 24, 3515–3540 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Li, X., Ma, S.: Choquard equations with critical nonlinearities. Commun. Contemp. Math. 22(04), 1950023 (2020)

    MathSciNet  MATH  Google Scholar 

  38. Li, G., Luo, X.: Existence and multiplicity of normalized solutions for a class of fractional Choquard equations. Sci. China Math. 63, 539–558 (2021)

    MathSciNet  MATH  Google Scholar 

  39. Li, G., Luo, X., Yang, T.: Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal perturbation. Math. Methods Appl. Sci. 44(13), 10331–10360 (2021)

    MathSciNet  MATH  Google Scholar 

  40. Lieb, E.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1976/1977)

  41. Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 147–174 (1987)

    MathSciNet  MATH  Google Scholar 

  42. Lieb, E., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  43. Lions, P.-L.: The Choquard equation and related questions. Nonlinear Anal. TMA 4, 1063–1072 (1980)

    MathSciNet  MATH  Google Scholar 

  44. Longhi, S.: Fractional Schrödinger equation in optics. Optics Lett. 40, 1117–1120 (2015)

    Google Scholar 

  45. Luo, H.: Ground state solutions of Pohozaev type for fractional Choquard equations with general nonlinearities. Comput. Math. Appl. 77, 877–887 (2019)

    MathSciNet  MATH  Google Scholar 

  46. Luo, H., Zhang, Z.: Normalized solutions to the fractional Schrödinger equations with combined nonlinearities. Calc. Var. Partial Differ. Equ. 59(4), 1–35 (2020)

    MATH  Google Scholar 

  47. Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)

    MathSciNet  MATH  Google Scholar 

  48. Ma, P., Zhang, J.: Existence and multiplicity of solutions for fractional Choquard equations. Nonlinear Anal. TMA 164, 100–117 (2017)

    MathSciNet  MATH  Google Scholar 

  49. Molica Bisci, G., Rădulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, vol. 162. Cambridge University Press, Cambridge (2016)

    MATH  Google Scholar 

  50. Moroz, V., van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)

    MathSciNet  MATH  Google Scholar 

  51. Moroz, V., van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)

    MathSciNet  MATH  Google Scholar 

  52. Moroz, I.M., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger-Newton equations Class. Quantum Grav. 15, 2733–2742 (1998). (Topology of the Universe Conf. (Cleveland, OH, 1997))

    MATH  Google Scholar 

  53. Mukherjee, T., Sreenadh, K.: Fractional Choquard equation with critical nonlinearities. Nonl. Differ. Equ. Appl. 24, 63, 34 (2017)

    MathSciNet  MATH  Google Scholar 

  54. Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28, 581–600 (1996)

    MathSciNet  MATH  Google Scholar 

  55. Penrose, R.: Quantum computation, entanglement and state reduction. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356, 1927–1939 (1998)

    MathSciNet  MATH  Google Scholar 

  56. Penrose, R.: The Road to Reality. Alfred A. Knopf Inc., New York (2005)

    MATH  Google Scholar 

  57. Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle. Akademie Verlag, Berlin (1954)

    MATH  Google Scholar 

  58. Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)

    MathSciNet  MATH  Google Scholar 

  59. Shen, Z., Gao, F., Yang, M.: Ground state for nonlinear fractional Choquard equations with general nonlinearities. Math. Methods Appl. Sci. 39, 4082–4098 (2016)

    MathSciNet  MATH  Google Scholar 

  60. Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)

    MathSciNet  MATH  Google Scholar 

  61. Soave, N.: Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case. J. Funct. Anal. 279, 108610, 43 (2020)

    MathSciNet  MATH  Google Scholar 

  62. Yang, T.: Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal \(L^2\)-critical or \(L^2\)-supercritical perturbation. J. Math. Phys. 61(5), 051505, 26 (2020)

    MATH  Google Scholar 

  63. Yang, Z., Zhao, F.: Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth. Adv. Nonlinear Anal. 10, 732–774 (2021)

    MathSciNet  MATH  Google Scholar 

  64. Ye, H.: Mass minimizers and concentration for nonlinear Choquard equations in \({\mathbb{R}}^{N},\) Topol. Methods Nonlinear Anal. 48(2), 393–417 (2016)

    MathSciNet  Google Scholar 

  65. Zhen, M., Zhang, B.: Normalized ground states for the critical fractional NLS equation with a perturbation. Rev. Mat. Complut. (2021). https://doi.org/10.1007/s13163-021-00388-w

    Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank two anonymous reviewers for the careful reading of the manuscript and for their valuable comments. Xiaoming He is supported by the National Natural Science Foundation of China (11771468, 11971027, 12171497), while Wenming Zou is supported by the National Natural Science Foundation of China (11771234, 11926323). The research of Vicenţiu D. Rădulescu is supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI-UEFISCDI, project number PCE 137/2021, within PNCDI III.

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Authors and Affiliations

  1. College of Science, Minzu University of China, Beijing, 100081, China

    Xiaoming He

  2. Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059, Kraków, Poland

    Vicenţiu D. Rădulescu

  3. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Vicenţiu D. Rădulescu

  4. China-Romania Research Center in Applied Mathematics, 200585, Craiova, Romania

    Vicenţiu D. Rădulescu

  5. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Wenming Zou

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  1. Xiaoming He
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  2. Vicenţiu D. Rădulescu
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Correspondence to Vicenţiu D. Rădulescu.

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He, X., Rădulescu, V.D. & Zou, W. Normalized Ground States for the Critical Fractional Choquard Equation with a Local Perturbation. J Geom Anal 32, 252 (2022). https://doi.org/10.1007/s12220-022-00980-6

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