Nondegeneracy of Ground States and Multiple Semiclassical Solutions of the Hartree Equation for General Dimensions


We study nondegeneracy of ground states of the Hartree equation

$$ -\Delta u+u=(I_{2}*u^2)u\quad \text{ in } {\mathbb {R}}^n $$

where \(n=3,4,5\) and \(I_2\) is the Newton potential. As an application of the nondegeneracy result, we use a Lyapunov–Schmidt reduction argument to construct multiple semiclassical solutions to the following Hartree equation with an external potential

$$ -\varepsilon ^2\Delta u+u+V(x)u=\varepsilon ^{-2}(I_{2}*u^2)u\quad \text{ in } {\mathbb {R}}^n. $$

This is a preview of subscription content, access via your institution.


  1. 1.

    Alves, C.O., Figueiredo, G.M., Yang, M.: Existence of solutions for a nonlinear choquard equation with potential vanishing at infinity. Adv. Nonlinear Anal. 5(4), 331–345 (2016)

    MathSciNet  Google Scholar 

  2. 2.

    Alves, C.O., Gao, F., Squassina, M., Yang, M.: Singularly perturbed critical Choquard equations. J. Differ. Equ. 263(7), 3943–3988 (2017)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Alves, C.O., Nóbrega, A.B., Yang, M.: Multi-bump solutions for Choquard equation with deepening potential well. Calc. Var. Partial Differ. Equ. 55(3), 1–28 (2016)

    MathSciNet  Google Scholar 

  4. 4.

    Alves, C.O., Yang, M.: Existence of semiclassical ground state solutions for a generalized Choquard equation. J. Differ. Equ. 257(11), 4133–4164 (2014)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Alves, C.O., Yang, M.: Multiplicity and concentration of solutions for a quasilinear Choquard equation. J. Math. Phys. 55(6), 061502 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ambrosetti, A., Badiale, M.: Variational perturbative methods and bifurcation of bound states from the essential spectrum. Proc. R. Soc. Edinb. Sect. A Math. 128, 1131–1161 (1998). 1

    MathSciNet  Article  Google Scholar 

  7. 7.

    Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 140(3), 285–300 (1997)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Ambrosetti, A., Malchiodi, A., Secchi, S.: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Rational Mech. Anal. 159(3), 253–271 (2001)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Amick, C.J., Toland, J.F.: Uniqueness and related analytic properties for the Benjamin-Ono equation–a nonlinear Neumann problem in the plane. Acta Math. 167(1–2), 107–126 (1991)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Bahrami, M., Großardt, A., Donadi, S., Bassi, A.: The Schrödinger-Newton equation and its foundations. New J. Phys. 16(11), 115007 (2014)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Bott, R.: Nondegenerate critical manifolds. Ann. Math. 60, 248–267 (1957)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 165(4), 295–316 (2002)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations, II. Calcul. Var. Partial Differ. Equ. 18(2), 207–219 (2003)

    Article  Google Scholar 

  14. 14.

    Chang, K.-C.: Infinite-dimensional Morse theory and multiple solution problems, volume 6 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Boston, MA (1993)

  15. 15.

    Chen, G.: Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations. Nonlinearity 28(4), 927 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Chen, G., Zheng, Y.: Concentration phenomenon for fractional nonlinear Schrödinger equations. Comm. Pure Appl. Anal. 13(6), 2359–2376 (2014)

    Article  Google Scholar 

  17. 17.

    Cingolani, S., Clapp, M., Secchi, S.: Multiple solutions to a magnetic nonlinear Choquard equation. Zeitschrift für Angew. Math. Phys. 63(2), 233–248 (2012)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Cingolani, S., Secchi, S.: Semiclassical analysis for pseudo-relativistic Hartree equations. J. Differ. Equ. 258(12), 4156–4179 (2015)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Cingolani, S., Secchi, S., Squassina, M.: Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc. R. Soc. Edinb. Sect. A Math. 140(5), 973–1009 (2010). 10

    Article  Google Scholar 

  20. 20.

    Cingolani, S., Tanaka, K.: Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well. arXiv preprint arXiv:1708.02356 (2017)

  21. 21.

    Dávila, J., del Pino, M., Dipierro, S., Valdinoci, E.: Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. Anal. PDE 8(5), 1165–1235 (2015)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Dávila, J., del Pino, M., Wei, J.: Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Differ. Equ. 256(2), 858–892 (2014)

    Article  Google Scholar 

  23. 23.

    del Pino, M., Felmer, P.L.: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149(1), 245–265 (1997)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Diósi, L.: Gravitation and quantum-mechanical localization of macro-objects. Phys. Lett. A 105(4), 199–202 (1984)

    Article  Google Scholar 

  25. 25.

    Frank, R.L., Lenzmann, E.: Uniqueness of non-linear ground states for fractional Laplacians in \({\mathbb{R}}\). Acta Math. 210(2), 261–318 (2013)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Frank, R.L., Lenzmann, E., Silvestre, L.: Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math. 69(9), 1671–1726 (2016)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Fröhlich, H.: Theory of electrical breakdown in ionic crytal. Proc. R. Soc. Ser. A 160(901), 230–241 (1937)

    Google Scholar 

  28. 28.

    Jones, K.R.W.: Gravitational self-energy as the litmus of reality. Mod. Phys. Lett. A 10(08), 657–667 (1995)

    Article  Google Scholar 

  29. 29.

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

    MathSciNet  Article  Google Scholar 

  30. 30.

    Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1976/77)

  31. 31.

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

  32. 32.

    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. the limit case, part I. Rev. Mat. Iber. 1(1), 145–201 (1985)

    Article  Google Scholar 

  33. 33.

    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. the limit case, part II. Rev. Mat. Iber. 1(2), 45–121 (1985)

    Article  Google Scholar 

  34. 34.

    Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Rational Mech. Anal. 195(2), 455–467 (2009)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Macrì, M., Nolasco, M.: Stationary solutions for the non-linear Hartree equation with a slowly varying potential. Nonlinear Differ. Equ. Appl. NoDEA 16(6), 681–715 (2009)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Moroz, I.M., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger-Newton equations. Classical Quantum Gravity 15(9), 2733–2742 (1998)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Moroz, V., Van Schaftingen, J.: Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials. Calcul. Var. Partial Differ. Equ. 37(1), 1 (2010)

    Article  Google Scholar 

  38. 38.

    Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265(2), 153–184 (2013). 7

    MathSciNet  Article  Google Scholar 

  39. 39.

    Moroz, V., Van Schaftingen, J.: Nonexistence and optimal decay of supersolutions to choquard equations in exterior domains. J. Differ. Equ. 254(8), 3089–3145 (2013)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Moroz, V., Van Schaftingen, J.: Semi-classical states for the Choquard equations. Calc. Var. Partial Differ. Equ. 52(1–2), 199–235 (2015)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Nolasco, M.: Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential. Commun. Pure Appl. Anal. 9(5), 1411–1419 (2010)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Pekar, S.: Untersuchung über die Elekronentheorie der Kristalle. Akedemie Verlag, Berlin (1954)

    Google Scholar 

  43. 43.

    Penrose, R.: On gravity’s role in quantum state reduction. Gener. Relat. Grav. 28(5), 581–600 (1996)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Penrose, R.: Quantum computation, entanglement and state reduction. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 356(1743), 1927–1939 (1998)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1975)

    Google Scholar 

  46. 46.

    Reed, M., Simon, B.: Methods of modern mathematical physics. II. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978)

    Google Scholar 

  47. 47.

    Secchi, S.: A note on Schrödinger-Newton systems with decaying electric potential. Nonlinear Anal. Theory Methods Appl. 72(9–10), 3842–3856 (2010)

    Article  Google Scholar 

  48. 48.

    Simon, B.: Harmonic Analysis. American Mathematical Society, Providence (2015)

    Google Scholar 

  49. 49.

    Tod, P., Moroz, I.M.: An analytical approach to the Schrödinger-Newton equations. Nonlinearity 12(2), 201–216 (1999)

    MathSciNet  Article  Google Scholar 

  50. 50.

    Wang, T., Yi, T.: Uniqueness of positive solutions of the choquard type equations. Appl. Anal. 96(3), 409–417 (2017)

    MathSciNet  Article  Google Scholar 

  51. 51.

    Wei, J., Winter, M.: Strongly interacting bumps for the Schrödinger-Newton equations. J. Math. Phys. 50(1), 012905 (2009). 22p

    MathSciNet  Article  Google Scholar 

  52. 52.

    Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3), 472–491 (1985)

    MathSciNet  Article  Google Scholar 

  53. 53.

    Xiang, C.-L.: Uniqueness and nondegeneracy of ground states for choquard equations in three dimensions. Calc. Var. Partial Differ. Equ. 55(6), 134 (2016)

    MathSciNet  Article  Google Scholar 

  54. 54.

    Yang, M., Ding, Y.: Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Commun. Pure Appl. Anal. 12(2), 771–783 (2013)

    MathSciNet  Article  Google Scholar 

  55. 55.

    Yang, M., Wei, Y.: Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities. J. Math. Anal. Appl. 403(2), 680–694 (2013)

    MathSciNet  Article  Google Scholar 

Download references


The author is indebted to Minbo Yang for many useful discussions on Choquard equation. This work is supported by the Zhejiang Provincial Natural Science Foundation of China (No. LY18A010023), NSFC (No.11771386) and First Class Discipline of Zhejiang - A (Zhejiang University of Finance and Economics- Statistics).

Author information



Corresponding author

Correspondence to Guoyuan Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, G. Nondegeneracy of Ground States and Multiple Semiclassical Solutions of the Hartree Equation for General Dimensions. Results Math 76, 34 (2021).

Download citation


  • Hartree equation
  • Schrödinger–Newton equation
  • nondegeneracy of ground states
  • semiclassical solutions
  • Lyapunov–Schmidt reduction

Mathematics Subject Classification

  • 35J61
  • 35J91
  • 35Q40