Milan Journal of Mathematics

, Volume 86, Issue 2, pp 329–342 | Cite as

Existence and Regularity of Solutions for a Choquard Equation with Zero Mass

  • Claudianor O. AlvesEmail author
  • Jianfu Yang


This paper concerns with the existence and regularity of solutions for the following Choquard type equation,
$$-\Delta_u = \big(I_{\mu} * F(u)\big) f(u) {\rm in} \mathbb{R}^3, \quad \quad (P)$$
where \({I_\mu = \frac{1}{|x|^\mu}, 0 < \mu < 3}\), is the Riesz potential, \({F(s)}\) is the primitive of the continuous function f(s), and \({I_{\mu} * F(u)}\) denotes the convolution of \({I_{\mu}}\) and F(u). By using the variational method, we prove that problem (P), in the zero mass case, possesses at least a nontrivial solution under certain conditions on f.

Mathematics Subject Classification (2010)

35J50 35J60 35A15 


Choquard equation nonlocal nonlinearities zero mass existence of solution regularity 


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  1. 1.
    Alves C.O., Montenegro M., Souto M.A.: Existence of a ground state solution for a nonlinear scalar field equation with critical growth. Calc. Var. Partial Differential Equations 43, 537–554 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alves C.O., Souto M.A.S.: Existence of solutions for a class of elliptic equations in \({\mathbb{R}^N}\) with vanishing potentials. J. Differential Equations 254, 1977–1991 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alves C.O., Figueiredo G.M., Yang M.: Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity. Advances in Nonlinear Analysis 5, 331–346 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ambrosetti A., Felli V., Malchiodi A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. 7, 117–144 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Azzollini and A. Pomponio, On a ”zero mass” nonlinear Schrödinger equation, Adv. Nonlinear Stud. 7 no. 4 (2007), 599-627.Google Scholar
  6. 6.
    A. Azzollini and A. Pomponio, Compactness results and applications to some ”zero mass” elliptic problems, Nonlinear Anal. 69 no. 10 (2008), 3559–3576.MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. Benci, C.R. Grisanti and A.M. Micheletti, Existence of solutions for the nonlinear Schrödinger equation with \({V (\infty) = 0}\), in: Progr. Nonlinear Differential Equations Appl., vol. 66, pp. 53–65, Birkhäuser, Basel, 2005Google Scholar
  8. 8.
    V. Benci, C.R. Grisanti and A.M. Micheletti, Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with \({V (\infty) = 0}\), Topol. Methods in Nonlinear Anal. 26 (2005), 203–219.Google Scholar
  9. 9.
    Berestycki H., Lions P.L.: Nonlinear scalar field equations, I existence of a ground state. Archive for Rational Mechanics and Analysis 82, 313–345 (1983)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Berestycki H., Gallouet T., Kavian O.: Equations de Champs scalaires euclidiens non linaires dans le plan. C.R. Acad. Sci. Paris Ser. I Math. 297, 307–310 (1984)zbMATHGoogle Scholar
  11. 11.
    Brezis H., Kato T.: Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. 58, 137–151 (1979)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cingolani S., Clapp M., Secchi S.: Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 63, 233–248 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    J.T. Devreese and A.S. Alexandrov, Advances in polaron physics, Springer Series in Solid-State Sciences, vol. 159, Springer, 2010.Google Scholar
  14. 14.
    L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on \({\mathbb{R}^N}\), Proc. Royal Soc. Edin. 129A (1999), 787–809.Google Scholar
  15. 15.
    Jones K.R.W.: Newtonian Quantum Gravity. Australian Journal of Physics 48, 1055–1081 (1995)CrossRefGoogle Scholar
  16. 16.
    Lenzmann E.: Uniqueness of ground states for pseudo-relativistic Hartree equations. Anal. PDE 2, 1–27 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57 (1976/77), 93–105.MathSciNetCrossRefGoogle Scholar
  18. 18.
    E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, AMS, Providence, Rhode island, 2001.Google Scholar
  19. 19.
    P.L. Lions, The concentration-compactness principle in the Calculus ov Variations. The Locally compact case, part 2, Anal. Inst. H. Poincaré Section C 1 (1984), 223–283.CrossRefGoogle Scholar
  20. 20.
    Lions P.L.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063–1072 (1980)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ma L., Zhao L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Menzala G.P.: On regular solutions of a nonlinear equation of Choquard’s type. Proc. Roy. Soc. Edinburgh Sect. A 86, 291–301 (1980)MathSciNetCrossRefGoogle Scholar
  23. 23.
    C. Mercuri, V. Moroz and J. Van Schaftingen, Groundstates and radial solutions to nonlinear Schrodinger-Poisson-Slater equations at the critical frequency. Calc. Var. Partial Differential Equations 55 (2016), 146.
  24. 24.
    V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equation: Hardy- Littlewood-Sobolev critical exponent, to appear in Commun. Contemp. Math., available at arXiv:1403.7414v1
  25. 25.
    Moroz V., Van Schaftingen J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Moroz V., Van Schaftingen J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Amer. Math. Soc. 367, 6557–6579 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Moroz I.M., Penrose R., Tod P.: Spherically-symmetric solution of the Schrödinger- Newton equation. Classical Quantum Gravity 15, 2733–2742 (1998)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pekar S.: Untersuchung über die Elektronentheorie der Kristalle. Akademie-Verlag, Berlin (1954)zbMATHGoogle Scholar
  29. 29.
    Secchi S.: A note on Schrödinger-Newton systems with decaying electric potential. Nonlinear Anal. 72, 3842–3856 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    M. Struwe, Variational Methods: Applications and to Partial Differential Equations and Hamiltonian systems, 4th ed., Springer, 2007.Google Scholar
  31. 31.
    Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helvetici 60, 558–581 (1985)MathSciNetCrossRefGoogle Scholar
  32. 32.
    J. Zhang and W. Zou, A Berestycki–Lions theorem revisited, Comm. Contemp. Math. 14 (2012), 1250033–1.MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Universidade Federal de Campina Grande Unidade Acadêmica de MatemáticaCampina GrandeBrazil
  2. 2.Department of MathematicsJiangxi Normal UniversityNanchang, JiangxiP.R. China

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