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A guide to the Choquard equation

  • Vitaly Moroz
  • Jean Van Schaftingen
Article

Abstract

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations
$$\begin{aligned} -\Delta u + V(x)u = \left( |x|^{-(N-\alpha )} *|u |^p\right) |u |^{p - 2} u \quad \text {in} \ \mathbb {R}^N, \end{aligned}$$
and some of its variants and extensions.

Keywords

Choquard equation Pekar polaron Schrödinger–Newton equation focusing Hartree equation attractive nonlocal interaction Riesz potential 

Mathematics Subject Classification

35Q55 (35R09, 35J91) 

References

  1. 1.
    Ackermann, N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248(2), 423–443 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ackermann, N.: A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations. J. Funct. Anal. 234(2), 277–320 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alves, C.O., Cassani, D., Tarsi, C., Yang, M.: Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in \({\mathbb{R}}^2\). J. Differ. Equ. 261(3), 1933–1972 (2016)zbMATHCrossRefGoogle Scholar
  4. 4.
    Alves, C.O., Figueiredo, G.M., Yang, M.: Multiple semiclassical solutions for a nonlinear Choquard equation with magnetic field. Asymptot. Anal. 96(2), 135–159 (2016)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Alves, C.O., Nóbrega, A.B., Yang, M.: Multi-bump solutions for Choquard equation with deepening potential well. Calc. Var. Partial Differ. Equ. 55(30, 55:48 (2016)Google Scholar
  6. 6.
    Alves, C.O., Yang, M.: Existence of semiclassical ground state solutions for a generalized Choquard equation. J. Differ. Equ. 257(11), 4133–4164 (2014)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Alves, C.O., Yang, M.: Multiplicity and concentration of solutions for a quasilinear Choquard equation. J. Math. Phys. 55, 061502 (2014)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ambrosetti, A.: On Schrödinger-Poisson systems. Milan J. Math. 76, 257–274 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ambrosetti, A., Garcia Azorero, J., Peral, I.: Perturbation of \(\Delta u+u^{(N+2)/(N-2)}=0\), the scalar curvature problem in \({\bf R}^N\), and related topics. J. Funct. Anal. 165(1), 117–149 (1999)Google Scholar
  10. 10.
    Ambrosetti, A., Ruiz, D.: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 10(3), 391–404 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Arriola, E.R., Soler, J.: A variational approach to the Schrödinger-Poisson system: asymptotic behaviour, breathers, and stability. J. Stat. Phys. 103(5–6), 1069–1105 (2001)zbMATHCrossRefGoogle Scholar
  13. 13.
    Aschbacher, W.H., Fröhlich, J., Graf, G.M., Schnee, K., Troyer, M.: Symmetry breaking regime in the nonlinear Hartree equation. J. Math. Phys. 43(8), 3879–3891 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Azzollini, A., d’Avenia, P., Luisi, V.: Generalized Schrödinger-Poisson type systems. Commun. Pure Appl. Anal. 12(2), 867–879 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Baernstein, A. II.: A unified approach to symmetrization. In: Partial differential equations of elliptic type (Cortona, 1992), Sympos. Math., vol. XXXV, pp. 47–91. Cambridge Univ. Press, Cambridge (1994)Google Scholar
  16. 16.
    Bahrami, M., Großardt, A., Donadi, S., Bassi, A.: The Schrödinger-Newton equation and its foundations. New J. Phys. 16, 115007 (2014)Google Scholar
  17. 17.
    Bartsch, T., Weth, T., Willem, M.: Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1–18 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Battaglia, L., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations in the plane. arXiv:1604.03294
  19. 19.
    Beckner, W.: Pitt’s inequality with sharp convolution estimates. Proc. Am. Math. Soc. 136(5), 1871–1885 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Beckner, W.: Pitt’s inequality and the fractional Laplacian: sharp error estimates. Forum Math. 24, 177–209 (2012)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Bellazzini, J., Frank, R.L., Visciglia, N.: Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems. Math. Ann. 360(3–4), 653–673 (2014)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Benci, V., Cerami, G.: Existence of positive solutions of the equation \(-\Delta u+a(x)u=u^{(N+2)/(N-2)}\) in \({ R}^N\). J. Funct. Anal. 88(1), 90–117 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11(2), 283–293 (1998)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Benguria, R., Brezis, H., Lieb, E.H.: The Thomas-Fermi-von Weizsäcker theory of atoms and molecules. Commun. Math. Phys. 79(2), 167–180 (1981)zbMATHCrossRefGoogle Scholar
  25. 25.
    Berestycki, H., Gallouët, T., Kavian, O.: Équations de champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris Sér. I Math. 297(5), 307–310 (1983)Google Scholar
  26. 26.
    Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Bernstein, D.H., Giladi, E., Jones, K.R.W.: Eigenstates of the gravitational Schrödinger equation. Mod. Phys. Lett. A 13(29), 2327–2336 (1998)CrossRefGoogle Scholar
  28. 28.
    Bonanno, C., d’Avenia, P., Ghimenti, M., Squassina, M.: Soliton dynamics for the generalized Choquard equation. J. Math. Anal. Appl. 417(1), 180–199 (2014)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Bonheure, D., Van Schaftingen, J.: Bound state solutions for a class of nonlinear Schrödinger equations. Rev. Mat. Iberoam. 24(1), 297–351 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Bongers, A.: Existenzaussagen für die Choquard-Gleichung: ein nichtlineares Eigenwertproblem der Plasma-Physik. Z. Angew. Math. Mech. 60(7), T240–T242 (1980)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Brascamp, H.J., Lieb, E.H., Luttinger, J.M.: A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17, 227–237 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Brezis, H., Kato, T.: Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. (9) 58(2), 137–151 (1979)Google Scholar
  33. 33.
    Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36(4), 437–477 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Brock, F., Solynin, AYu.: An approach to symmetrization via polarization. Trans. Am. Math. Soc. 352(4), 1759–1796 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Brothers, J.E., Ziemer, W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Buffoni, B., Jeanjean, L., Stuart, C.A.: Existence of a nontrivial solution to a strongly indefinite semilinear equation. Proc. Am. Math. Soc. 119(1), 179–186 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Burchard, A.: Cases of equality in the Riesz rearrangement inequality. Ann. Math. (2) 143(3), 499–527 (1996)Google Scholar
  39. 39.
    Byeon, J., Jeanjean, L.: Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete Contin. Dyn. Syst. 19(2), 255–269 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Byeon, J., Jeanjean, L.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185(2), 185–200 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. II. Calc. Var. Partial Differ. Equ. 18(2), 207–219 (2003)zbMATHCrossRefGoogle Scholar
  42. 42.
    Cao, D.M.: The existence of nontrivial solutions to a generalized Choquard-Pekar equation. Acta Math. Sci. (Chinese) 9(1), 101–112 (1989). (Chinese)MathSciNetGoogle Scholar
  43. 43.
    Cao, P., Wang, J., Zou, W.: On the standing waves for nonlinear Hartree equation with confining potential. J. Math. Phys. 53(3), 033702, 27 (2012)Google Scholar
  44. 44.
    Castro, A., Cossio, J., Neuberger, J.M.: A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems. Electron. J. Differ. Equ. 1998(2), 18 (1998)Google Scholar
  45. 45.
    Castro, A., Cossio, J., Neuberger, J.M.: A sign-changing solution for a superlinear Dirichlet problem. Rocky Mt. J. Math. 27(4), 1041–1053 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Catto, I., Dolbeault, J., Sánchez, O., Soler, J.: Existence of steady states for the Maxwell-Schrödinger-Poisson system: exploring the applicability of the concentrationcompactness principle. Math. Models Methods Appl. Sci. 23(10), 1915–1938 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Catto, I., Le Bris, C., Lions, P.-L.: On some periodic Hartree-type models for crystals. Ann. Inst. H. Poincaré Anal. Non Linéaire 19(2), 143–190 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Cerami, G., Solimini, S., Struwe, M.: Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal. 69(3), 289–306 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Chen, J., Guo, B.: Blow up solutions for one class of system of Pekar-Choquard type nonlinear Schrödinger equation. Appl. Math. Comput. 186(1), 83–92 (2007)zbMATHMathSciNetGoogle Scholar
  50. 50.
    Chen, S., Xiao, L.: Existence of a nontrivial solution for a strongly indefinite periodic Choquard system. Calc. Var. Partial Differ. Equ. 54(1), 599–614 (2015)zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Chen, W., Li, C., Ou, B.: Qualitative properties of solutions for an integral equation. Discrete Contin. Dyn. Syst. 12(2), 347–354 (2005)zbMATHMathSciNetGoogle Scholar
  52. 52.
    Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59(3), 330–343 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Chen, H., Zhou, F.: Classification of isolated singularities of positive solutions for Choquard equations. J. Differ. Equ. doi: 10.1016/j.jde.2016.08.047
  54. 54.
    Choquard, P., Stubbe, J.: The one-dimensional Schrödinger-Newton equations. Lett. Math. Phys. 81(2), 177–184 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Choquard, P., Stubbe, J., Vuffray, M.: Stationary solutions of the Schrödinger- Newton model—an ODE approach. Differ. Integral Equ. 21(7–8), 665–679 (2008)zbMATHGoogle Scholar
  56. 56.
    Cingolani, S., Clapp, M., Secchi, S.: Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 63(2), 233–248 (2012)zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Cingolani, S., Clapp, M., Secchi, S.: Intertwining semiclassical solutions to a Schrödinger-Newton system. Discrete Contin. Dyn. Syst. Ser. S 6(4), 891–908 (2013)zbMATHMathSciNetGoogle Scholar
  58. 58.
    Cingolani, S., Secchi, S.: Multiple S1-orbits for the Schrödinger-Newton system. Differ. Integral Equ. 26(9/10), 867–884 (2013)zbMATHGoogle Scholar
  59. 59.
    Cingolani, S., Secchi, S.: Ground states for the pseudo-relativistic Hartree equation with external potential. Proc. R. Soc. Edinb. Sect. A 145(1), 73–90 (2015)zbMATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    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 140(5), 973–1009 (2010)zbMATHCrossRefGoogle Scholar
  61. 61.
    Cingolani, S., Weth, T.: On the planar Schrödinger-Poisson system. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(1), 169–197 (2016)zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Clapp, M., Salazar, D.: Positive and sign changing solutions to a nonlinear Choquard equation. J. Math. Anal. Appl. 407(1), 1–15 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    Coti Zelati, V., Nolasco, M.: Existence of ground states for nonlinear, pseudorelativistic Schrödinger equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22(1), 51–72 (2011)Google Scholar
  64. 64.
    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, p. 18. Springer, Berlin (1987)Google Scholar
  65. 65.
    d’Avenia, P., Squassina, M.: Soliton dynamics for the Schrödinger-Newton system. Math. Models Methods Appl. Sci. 24(3), 553–572 (2014)zbMATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    d’Avenia, P., Siciliano, G., Squassina, M.: On fractional Choquard equations. Math. Models Methods Appl. Sci. 25(8), 1447–1476 (2015)zbMATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Deng, Y., Lu, L., Shuai, W.: Constraint minimizers of mass critical Hartree energy functionals: Existence and mass concentration. J. Math. Phys. 56(6), 061503, 15 (2015)Google Scholar
  68. 68.
    del Pino, M., Felmer, P.L.: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149(1), 245–265 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    De Napoli, P.L., Drelichman, I.: Elementary proofs of embedding theorems for potential spaces of radial functions. In: Ruzhansky, M., Tikhonov, S. (eds.) Methods of Fourier Analysis and Approximation Theory, pp. 115–138. Birkhäuser, Basel (2016)Google Scholar
  70. 70.
    Di Cosmo, J., Van Schaftingen, J.: Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field. J. Differ. Equ. 259(2), 596–627 (2015)zbMATHCrossRefGoogle Scholar
  71. 71.
    Diósi, L.: Gravitation and quantum-mechanical localization of macro-objects. Phys. Lett. A 105(4–5), 199–202 (1984)CrossRefGoogle Scholar
  72. 72.
    Disconzi, M.M.: A priori estimates for a critical Schrödinger-Newton equation. In: Proceedings of the Ninth MSU-UAB Conference on Differential Equations and Computational Simulations, Electron. J. Differ. Equ. Conf., vol. 20, pp. 39–51, Texas State Univ., San Marcos (2013)Google Scholar
  73. 73.
    Donsker, M.D., Varadhan, S.R.S.: The polaron problem and large deviations. Phys. Rep. 77(3), 235–237 (1981, New stochasitic methods in physics)Google Scholar
  74. 74.
    Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the polaron. Commun. Pure Appl. Math. 36(4), 505–528 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    Duoandikoetxea, J.: Fractional integrals on radial functions with applications to weighted inequalities. Ann. Mat. Pura Appl. (4) 192(4), 553–568 (2013)Google Scholar
  76. 76.
    du Plessis, N.: Some theorems about the Riesz fractional integral. Trans. Am. Math. Soc. 80, 124–134 (1955)zbMATHCrossRefMathSciNetGoogle Scholar
  77. 77.
    du Plessis, N.: An introduction to potential theory, University Mathematical Monographs, No. 7. Hafner Publishing Co., Darien, Conn., Oliver and Boyd, Edinburgh (1970)Google Scholar
  78. 78.
    Dymarskii, Ya.M.: The periodic Choquard equation. In: Differential operators and related topics, Vol. I, Oper. Theory Adv. Appl., vol. 117, pp. 87–99. Birkhäuser, Basel (2000)Google Scholar
  79. 79.
    Efinger, H.J.: On the theory of certain nonlinear Schrödinger equations with nonlocal interaction. Nuovo Cimento B (11) 80(2), 260–278 (1984)Google Scholar
  80. 80.
    Feng, B.: Sharp threshold of global existence and instability of standing wave for the Schrödinger-Hartree equation with a harmonic potential. Nonlinear Anal. Real World Appl. 31, 132–145 (2016)zbMATHCrossRefMathSciNetGoogle Scholar
  81. 81.
    Frank, R.L., Geisinger, L.: The ground state energy of a polaron in a strong magnetic field. Commun. Math. Phys. 338(1), 1–29 (2015)zbMATHCrossRefMathSciNetGoogle Scholar
  82. 82.
    Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69(3), 397–408 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  83. 83.
    Frank, R.L., Lenzmann, E.: On ground states for the L2-critical boson star equation. arXiv:0910.2721
  84. 84.
    Frank, R.L., Lieb, E.H.: Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality. Calc. Var. Partial Differ. Equ. 39(1–2), 85–99 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  85. 85.
    Franklin, J., Guo, Y., McNutt, A., Morgan, A.: The Schrödinger-Newton system with self-field coupling. Class. Quantum Gravity 32(6), 065010 (2015)Google Scholar
  86. 86.
    Fröhlich, H.: Theory of electrical breakdown in ionic crystal. Proc. R. Soc. Ser. A 160(901), 230–241 (1937)CrossRefGoogle Scholar
  87. 87.
    Fröhlich, H.: Electrons in lattice fields. Adv. Phys. 3(11) (1954)Google Scholar
  88. 88.
    Fröhlich, J., Lenzmann, E.: Mean-field limit of quantum Bose gases and nonlinear Hartree equation. In: Séminaire: Équations aux Dérivées Partielles (2003), Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, pp. Exp. No. XIX, 26 (2004)Google Scholar
  89. 89.
    Fröhlich, J., Jonsson, B.L.G., Lenzmann, E.: Boson stars as solitary waves. Commun. Math. Phys. 274(1), 1–30 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  90. 90.
    Gao, F., Yang, M.: On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation. arXiv:1604.00826
  91. 91.
    Gao, F., Yang, M.: Existence and multiplicity of solutions for a class of Choquard equations with Hardy-Littlewood-Sobolev critical exponent. arXiv:1605.05038
  92. 92.
    Genev, H., Venkov, G.: Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete Contin. Dyn. Syst. Ser. S 5(5), 903–923 (2012)zbMATHCrossRefMathSciNetGoogle Scholar
  93. 93.
    Ghergu, M., Taliaferro, S.D.: Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity. J. Differ. Equ. 261(1), 189–217 (2016)zbMATHCrossRefMathSciNetGoogle Scholar
  94. 94.
    Ghimenti, M., Moroz, V., Van Schaftingen, J.: Least action nodal solutions for the quadratic Choquard equation. Proc. Am. Math. Soc. doi: 10.1090/proc/13247
  95. 95.
    Ghimenti, M., Van Schaftingen, J.: Nodal solutions for the Choquard equation. J. Funct. Anal. 271(1), 107–135 (2016)zbMATHCrossRefMathSciNetGoogle Scholar
  96. 96.
    Ghoussoub, N.: Self-dual partial differential systems and their variational principles, Springer Monographs in Mathematics. Springer, New York (2009)Google Scholar
  97. 97.
    Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  98. 98.
    Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations with nonlocal interaction. Math. Z. 170(2), 109–136 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  99. 99.
    Giulini, D., Großardt, A.: The Schrödinger-Newton equation as a non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields. Class. Quantum Gravity 29(21), 215010, 25 (2012)Google Scholar
  100. 100.
    Griesemer, M., Hantsch, F., Wellig, D.: On the magnetic Pekar functional and the existence of bipolarons. Rev. Math. Phys. 24(6), 1250014, 13 (2012)Google Scholar
  101. 101.
    Guo, Q., Su, Y.: Instability of standing waves for inhomogeneous Hartree equations. J. Math. Anal. Appl. 437(2), 1159–1175 (2016)zbMATHCrossRefMathSciNetGoogle Scholar
  102. 102.
    Gustafson, K., Sather, D.: A branching analysis of the Hartree equation. Rend. Mat. (6) 4, 723–734 (1971)Google Scholar
  103. 103.
    Guzmán, F.S., Ureña-López, L.A.: Newtonian collapse of scalar field dark matter. Phys. Rev. D 68, 024023 (2003)CrossRefGoogle Scholar
  104. 104.
    Guzmán, F.S., Ureña-López, L.A.: Evolution of the Schrödinger-Newton system for a self-gravitating scalar field. Phys. Rev. D 69, 124033 (2004)CrossRefGoogle Scholar
  105. 105.
    Hajaiej, H.: Schrödinger systems arising in nonlinear optics and quantum mechanics. Part I. Math. Models Methods Appl. Sci. 22(7), 1250010, 27 (2012)Google Scholar
  106. 106.
    Hajaiej, H.: On Schrödinger systems arising in nonlinear optics and quantum mechanics: II. Nonlinearity 26(4), 959–970 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  107. 107.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press (1952)Google Scholar
  108. 108.
    Herbst, I.W.: Spectral theory of the operator \((p^{2}+m^{2})^{1/2}-Ze^{2}/r\). Commun. Math. Phys. 53(3), 285–294 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  109. 109.
    Hirata, J., Ikoma, N., Tanaka, K.: Nonlinear scalar field equations in RN: mountainpass and symmetric mountain-pass approaches. Topol. Methods Nonlinear Anal. 35(2), 253–276 (2010)zbMATHMathSciNetGoogle Scholar
  110. 110.
    Ianni, I., Ruiz, D.: Ground and bound states for a static Schrödinger-Poisson-Slater problem. Commun. Contemp. Math. 14(1), 1250003, 22 (2012)Google Scholar
  111. 111.
    Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28(10), 1633–1659 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  112. 112.
    Jeong, W., Seok, J.: On perturbation of a functional with the mountain-pass geometry: applications to the nonlinear Schrödinger-Poisson equations and the nonlinear Klein-Gordon-Maxwell equations. Calc. Var. Partial Differ. Equ. 49(1–2), 649–668 (2014)zbMATHCrossRefGoogle Scholar
  113. 113.
    Jones, K.R.W.: Gravitational self-energy as the litmus of reality. Mod. Phys. Lett. A 10(8), 657–668 (1995)CrossRefGoogle Scholar
  114. 114.
    Jones, K.R.W.: Newtonian quantum gravity. Aust. J. Phys. 48(6), 1055–1081 (1995)CrossRefGoogle Scholar
  115. 115.
    Karasev, M.V., Maslov, V.P.: Quasiclassical soliton solutions of the Hartree equation. Newtonian interaction with screening, Teoret. Mat. Fiz. 40(2), 235–244 (1979) (Russian); English transl., Theoret. and Math. Phys. 40(2), 715–721 (1979)Google Scholar
  116. 116.
    Kawohl, B.: Rearrangements and convexity of level sets in PDE. Lecture Notes in Mathematics, vol. 1150. Springer, Berlin (1985)Google Scholar
  117. 117.
    Krasnosel’skiĭ, M.A.: Topological methods in the theory of nonlinear integral equations, translated by A. H. Armstrong, p. 6. MacMillan, New York (1964)Google Scholar
  118. 118.
    Kumar, D., Soni, V.: Single particle Schrödinger equation with gravitational selfinteraction. Phys. Lett. A 271(3), 157–166 (2000)CrossRefMathSciNetGoogle Scholar
  119. 119.
    Küpper, T., Zhang, Z., Xia, H.: Multiple positive solutions and bifurcation for an equation related to Choquard’s equation. Proc. Edinb. Math. Soc. (2) 46(3), 597–607 (2003)Google Scholar
  120. 120.
    Landkof, N.S.: Foundations of modern potential theory, translated by A. P. Doohovskoy, Grundlehren der mathematischen Wissenschaften, Springer, New York–Heidelberg (1972)Google Scholar
  121. 121.
    Le Bris, C., Lions, P.-L.: From atoms to crystals: a mathematical journey. Bull. Amer. Math. Soc. (N.S.) 42(3), 291–363 (2005)Google Scholar
  122. 122.
    Lei, Y.: On the regularity of positive solutions of a class of Choquard type equations. Math. Z. 273(3–4), 883–905 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  123. 123.
    Lei, Y.: Qualitative analysis for the static Hartree-type equations. SIAM J. Math. Anal. 45(1), 388–406 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  124. 124.
    Lenzmann, E.: Uniqueness of ground states for pseudorelativistic Hartree equations. Anal. PDE 2(1), 1–27 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  125. 125.
    Lewin, M., Nam, P.T., Rougerie, N.: Derivation of Hartree’s theory for generic mean-field Bose systems. Adv. Math. 254, 570–621 (2014)zbMATHCrossRefMathSciNetGoogle Scholar
  126. 126.
    Li, G.-B., Ye, H.-Y.: The existence of positive solutions with prescribed L2-norm for nonlinear Choquard equations. J. Math. Phys. 55(12), 121501, 19 (2014)Google Scholar
  127. 127.
    Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1976/77)Google Scholar
  128. 128.
    Lieb, E.H.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. (2) 118(2), 349–374 (1983)Google Scholar
  129. 129.
    Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence (2001)Google Scholar
  130. 130.
    Lieb, E.H., Thomas, L.E.: Exact ground state energy of the strong-coupling polaron. Commun. Math. Phys. 183(3), 511–519 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  131. 131.
    Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112(1), 147–174 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  132. 132.
    Lions, P.-L.: The Choquard equation and related questions. Nonlinear Anal. 4(6), 1063–1072 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  133. 133.
    Lions, P.-L.: Compactness and topological methods for some nonlinear variational problems of mathematical physics, Nonlinear problems: present and future (Los Alamos, N.M., 1981), North-Holland Math. Stud., vol. 61, pp. 17–34. North-Holland, Amsterdam–New York (1982)Google Scholar
  134. 134.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(2), 109–145 (1984)Google Scholar
  135. 135.
    Lions, P.-L.: Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109(1), 33–97 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  136. 136.
    Lopes, O., Maris, M.: Symmetry of minimizers for some nonlocal variational problems. J. Funct. Anal. 254(2), 535–592 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  137. 137.
    Lü, D.: Existence and concentration of solutions for a nonlinear Choquard equation. Mediterr. J. Math. 12(3), 839–850 (2015)zbMATHCrossRefMathSciNetGoogle Scholar
  138. 138.
    Macrì, M., Nolasco, M.: Stationary solutions for the non-linear Hartree equation with a slowly varying potential. NoDEA Nonlinear Differ. Equ. Appl. 16(6), 681–715 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  139. 139.
    Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195(2), 455–467 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  140. 140.
    Maeda, M., Masaki, S.: An example of stable excited state on nonlinear Schrödinger equation with nonlocal nonlinearity. Differ. Integral Equ. 26(7–8), 731–756 (2013)zbMATHGoogle Scholar
  141. 141.
    Manfredi, G.: The Schrödinger-Newton equations beyond Newton. Gen. Relativ. Gravity 47(2), 1 (2015)Google Scholar
  142. 142.
    Mawhin, J., Willem, M.: Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol. 74, p. 6. Springer, New York (1989)CrossRefGoogle Scholar
  143. 143.
    Menzala, G.P.: On regular solutions of a nonlinear equation of Choquard’s type. Proc. R. Soc. Edinb. Sect. A 86(3–4), 291–301 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  144. 144.
    Menzala, G.P.: On the nonexistence of solutions for an elliptic problem in unbounded domains. Funkcial. Ekvac. 26(3), 231–235 (1983)zbMATHMathSciNetGoogle Scholar
  145. 145.
    Mercuri, C., Moroz, V., Van Schaftingen, J.: Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency. Calc. Var. Partial Differ. Equ. arXiv:1507.02837
  146. 146.
    Møller, C., The energy-momentum complex in general relativity and related problems. In: Les théories relativistes de la gravitation (Royaumont, 1959), pp. 15–29. Éditions du Centre National de la Recherche Scientifique, Paris (1959)Google Scholar
  147. 147.
    Moroz, I.M., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger-Newton equations. Class. Quantum Gravity 15(9), 2733–2742 (1998)zbMATHCrossRefGoogle Scholar
  148. 148.
    Moroz, V., Van Schaftingen, J.: Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials. Calc. Var. Partial Differ. Equ. 37(1–2), 1–27 (2010)zbMATHCrossRefGoogle Scholar
  149. 149.
    Moroz, V., Van Schaftingen, J.: Nonlocal Hardy type inequalities with optimal constants and remainder terms. Ann. Univ. Buchar. Math. Ser. 3 (LXI)(2), 187–200 (2012)Google Scholar
  150. 150.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  151. 151.
    Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  152. 152.
    Moroz, V., Van Schaftingen, J.: Semi-classical states for the Choquard equation. Calc. Var. Partial Differ. Equ. 52(1–2), 199–235 (2015)zbMATHCrossRefMathSciNetGoogle Scholar
  153. 153.
    Moroz, V., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367(9), 6557–6579 (2015)zbMATHCrossRefMathSciNetGoogle Scholar
  154. 154.
    Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent. Commun. Contemp. Math. 17(5), 1550005 (12 pages) (2015)Google Scholar
  155. 155.
    Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261–274 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  156. 156.
    Mugnai, D.: The Schrödinger-Poisson system with positive potential. Commun. Partial Differ. Equ. 36(7), 1099–1117 (2011)zbMATHCrossRefGoogle Scholar
  157. 157.
    Mukherjee, T., Sreenadh, K.: Existence and multiplicity results for Brezis-Nirenberg type fractional Choquard equation. arXiv:1605.06805
  158. 158.
    Mugnai, D.: Pseudorelativistic Hartree equation with general nonlinearity: existence, non-existence and variational identities. Adv. Nonlinear Stud. 13(4), 799–823 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  159. 159.
    Nolasco, M.: Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential. Commun. Pure Appl. Anal. 9(5), 1411–1419 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  160. 160.
    Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle, p. 2. Akademie Verlag, Berlin (1954)zbMATHGoogle Scholar
  161. 161.
    Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relat. Gravit. 28(5), 581–600 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  162. 162.
    Penrose, R.: Quantum computation, entanglement and state reduction. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356(1743), 1927–1939 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  163. 163.
    Pinchover, Y., Tintarev, K.: A ground state alternative for singular Schrödinger operators. J. Funct. Anal. 230(1), 65–77 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  164. 164.
    Pohožaev, S.I.: On the eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\). Dokl. Akad. Nauk SSSR 165, 36–39 (1965) (Russian); English transl., Soviet Math. Dokl. 6, 1408–1411 (1965)Google Scholar
  165. 165.
    Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics. In: Annals of Mathematics Studies, no. 27. Princeton University Press, Princeton (1951)Google Scholar
  166. 166.
    Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35(3), 681–703 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  167. 167.
    Quittner, P., Souplet, P.: Superlinear parabolic problems: Blow-up, global existence and steady states, Birkhäuser Advanced Texts: Basler Lehrbücher, p. 17. Birkhäuser, Basel (2007)Google Scholar
  168. 168.
    Rabinowitz, P.H.: Variational methods for nonlinear eigenvalue problems. In: Prodi, G. (ed.) Eigenvalues of non-linear problems (Varenna, 1974), 139–195. Edizioni Cremonese, Rome (1974)Google Scholar
  169. 169.
    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence (1986)CrossRefGoogle Scholar
  170. 170.
    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  171. 171.
    Rabinowitz, P.H.: Critical point theory and applications to differential equations: a survey. In: Topological nonlinear analysis, Progr. Nonlinear Differential Equations Appl., vol. 15, pp. 464–513. Birkhäuser Boston, Boston (1995)Google Scholar
  172. 172.
    Riesz, F.: Sur une inégalité intégrale. J. Lond. Math. Soc. S1-5(3), 162 (1930)Google Scholar
  173. 173.
    Riesz, M.: L’intégrale de Riemann-Liouville et le problème de Cauchy pour l’équation des ondes. Bull. Soc. Math. France 67, 153–170 (1939)zbMATHCrossRefMathSciNetGoogle Scholar
  174. 174.
    Riesz, M.: L’intégrale de Riemann-Liouville et le problème de Cauchy. Acta Math. 81, 1–223 (1949)zbMATHCrossRefMathSciNetGoogle Scholar
  175. 175.
    Rosenfeld, L.: On quantization of fields. Nucl. Phys. 40, 353–356 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  176. 176.
    Rubin, B.S.: One-dimensional representation, inversion and certain properties of Riesz potentials of radial functions. Mat. Zametki 34(4), 521–533 (1983). (Russian)MathSciNetGoogle Scholar
  177. 177.
    Ruffini, R., Bonazzola, S.: Systems of self-gravitating particles in general relativity and the concept of an equation of state. Phys. Rev. 187(5), 1767–1783 (1969)CrossRefGoogle Scholar
  178. 178.
    Ruiz, D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237(2), 655–674 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  179. 179.
    Ruiz, D.: On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases. Arch. Ration. Mech. Anal. 198(1), 349–368 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  180. 180.
    Ruiz, D., Van Schaftingen, J.: Odd symmetry of least energy nodal solutions for the Choquard equation (2016). arXiv:1606.05668
  181. 181.
    Salazar, D.: Vortex-type solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 66(3), 663–675 (2015)zbMATHCrossRefMathSciNetGoogle Scholar
  182. 182.
    Samko, S.G.: Hypersingular integrals and their applications, Analytical Methods and Special Functions, vol. 5. Taylor & Francis Ltd, London (2002)zbMATHGoogle Scholar
  183. 183.
    Samko, S.: Best constant in the weighted Hardy inequality: the spatial and spherical version. Fract. Calc. Appl. Anal. 8(1), 39–52 (2005)zbMATHMathSciNetGoogle Scholar
  184. 184.
    Schunck, F.E., Mielke, E.W.: General relativistic boson stars. Class. Quantum Gravity 20(20), R301–R356 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  185. 185.
    Secchi, S.: A note on Schrödinger-Newton systems with decaying electric potential. Nonlinear Anal. 72(9–10), 3842–3856 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  186. 186.
    Siegel, D., Talvila, E.: Pointwise growth estimates of the Riesz potential. In: Dynam. Contin. Discrete Impuls. Systems 5(1–4), 185–194 (1999, Differential equations and dynamical systems (Waterloo, ON, 1997))Google Scholar
  187. 187.
    Sobolev, S.L.: On a theorem of functional analysis. Math. Sbornik 4 46(3), 5–9 (1938) (Russian); English transl., Amer. Math. Soc. Transl. 2 (1938), 39–68Google Scholar
  188. 188.
    Souto,M.A.S., de Lima, R.N.: Choquard equations with mixed potential. arXiv:1506.08179
  189. 189.
    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)Google Scholar
  190. 190.
    Stein, E.M., Weiss, G.: Fractional integrals on \(n\)-dimensional Euclidean space. J. Math. Mech. 7, 503–514 (1958)zbMATHMathSciNetGoogle Scholar
  191. 191.
    Stein, E.M., Zygmund, A.: Boundedness of translation invariant operators on Hölder spaces and \(L^{p}\)-spaces. Ann. Math. (2) 85, 337–349 (1967)Google Scholar
  192. 192.
    Struwe, M.: Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems, 4th edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34. Springer, Berlin (2008)Google Scholar
  193. 193.
    Stuart, C.A.: Bifurcation for variational problems when the linearisation has no eigenvalues. J. Funct. Anal. 38(2), 169–187 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  194. 194.
    Stubbe, J.: Bound states of two-dimensional Schrödinger-Newton equations. arXiv:0807.4059
  195. 195.
    Stubbe, J., Vuffray, M.: Bound states of the Schrödinger-Newton model in low dimensions. Nonlinear Anal. 73(10), 3171–3178 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  196. 196.
    Sun, X., Zhang, Y.: Multi-peak solution for nonlinear magnetic Choquard type equation. J. Math. Phys. 55(3), 031508 (25 p.) (2014)Google Scholar
  197. 197.
    Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 9(3), 281–304 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  198. 198.
    Tod, K.P.: The ground state energy of the Schrödinger-Newton equation. Phys. Lett. A 280(4), 173–176 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  199. 199.
    Thim, J.: Asymptotics and inversion of Riesz potentials through decomposition in radial and spherical parts. Ann. Mat. Pura Appl. (4) 195(2), 323–341 (2016)Google Scholar
  200. 200.
    Tod, K.P., Moroz, I.M.: An analytical approach to the Schrödinger-Newton equations. Nonlinearity 12(2), 201–216 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  201. 201.
    Trudinger, N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3) 22, 265–274 (1968)Google Scholar
  202. 202.
    Vaira, G.: Ground states for Schrödinger-Poisson type systems. Ric. Mat. 60(2), 263–297 (2011)zbMATHCrossRefMathSciNetGoogle Scholar
  203. 203.
    Vaira, G.: Existence of bound states for Schrödinger-Newton type systems. Adv. Nonlinear Stud. 13(2), 495–516 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  204. 204.
    Van Schaftingen, J., Willem, M.: Symmetry of solutions of semilinear elliptic problems. J. Eur. Math. Soc. (JEMS) 10(2), 439–456 (2008)zbMATHMathSciNetGoogle Scholar
  205. 205.
    Van Schaftingen, J., Xia, J.: Solutions to the Choquard equation under coercive potentials. arXiv:1607.00151
  206. 206.
    Wang, T.: Existence and nonexistence of nontrivial solutions for Choquard type equations. Electron. J. Diff. Equ. 2016(03), 1–17 (2016)zbMATHMathSciNetGoogle Scholar
  207. 207.
    Wang, T., Yia, T.: Uniqueness of positive solutions of the Choquard type equations. Appl. Anal. doi: 10.1080/00036811.2016.1138473
  208. 208.
    Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87(4), 567–576 (1982/83)Google Scholar
  209. 209.
    Wei, J., Winter, M.: Strongly interacting bumps for the Schrödinger-Newton equations. J. Math. Phys. 50(1), 012905 (22 p.) (2009)Google Scholar
  210. 210.
    Weth, T.: Spectral and variational characterizations of solutions to semilinear eigenvalue problems, doctoral dissertation, p. 6. Johannes Gutenberg-Universität, Mainz (2001)Google Scholar
  211. 211.
    Willem, M.: Minimax theorems. In: Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser, Boston, Mass (1996)Google Scholar
  212. 212.
    Xiang, C.-L.: Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions. arXiv:1506.01550
  213. 213.
    Xie, T., Xiao, L., Wang, J.: Existence of multiple positive solutions for Choquard equation with perturbation. Adv. Math. Phys. 760157 (10 p.) (2015)Google Scholar
  214. 214.
    Yafaev, D.: Sharp constants in the Hardy-Rellich inequalities. J. Funct. Anal. 168(1), 121–144 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  215. 215.
    Yang, M., Ding, Y.: Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Commun. Pure Appl. Anal. 12(2), 771–783 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  216. 216.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  217. 217.
    Yang, M., Zhang, J., Zhang, Y.: Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity. arXiv:1604.04715
  218. 218.
    Ye, H.-Y.: The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in \({\mathbb{R}^N}\). J. Math. Anal. Appl. 431(2), 935–954 (2015)zbMATHCrossRefMathSciNetGoogle Scholar
  219. 219.
    Ye, H.-Y.: Mass minimizers and concentration for nonlinear Choquard equations in \({\mathbb{R}}^N\). arXiv:1502.01560
  220. 220.
    Z. Zhang, Multiple solutions of the Choquard equation. In: Differential equations and control theory (Wuhan: 1994), Lecture Notes in Pure and Appl. Math., vol. 176, pp. 477–482. Dekker, New York (1996)Google Scholar
  221. 221.
    Zhang, Z.: Multiple solutions of nonhomogeneous for related Choquard’s equation. Acta Math. Sci. Ser. B Engl. Ed. 20(3), 374–379 (2000)zbMATHMathSciNetGoogle Scholar
  222. 222.
    Zhang, Z.: Multiple solutions of nonhomogeneous Chouquard’s equations. Acta Math. Appl. Sin. (English Ser.) 17(1), 47–52 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  223. 223.
    Zhang, Z., Küpper, T., Hu, A., Xia, H.: Existence of a nontrivial solution for Choquard’s equation. Acta Math. Sci. Ser. B Engl. Ed. 26(3), 460–468 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  224. 224.
    Zhao, L., Zhao, F.: On the existence of solutions for the Schrödinger-Poisson equations. J. Math. Anal. Appl. 346(1), 155–169 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  225. 225.
    Zhao, L., Zhao, F., Shi, J.: Higher dimensional solitary waves generated by secondharmonic generation in quadratic media. Calc. Var. Partial Differ. Equ. 54(3), 2657–2691 (2015)zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsSwansea UniversitySwanseaWales, UK
  2. 2.Institut de Recherche en Mathématique et PhysiqueUniversité catholique de LouvainLouvain-la-NeuveBelgium

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