Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency

  • Carlo Mercuri
  • Vitaly Moroz
  • Jean Van Schaftingen


We study the nonlocal Schrödinger–Poisson–Slater type equation
$$\begin{aligned} - \Delta u + (I_\alpha *\vert u\vert ^p)\vert u\vert ^{p - 2} u= \vert u\vert ^{q-2}u\quad \text {in}\quad \mathbb {R}^N, \end{aligned}$$
where \(N\in \mathbb {N}\), \(p>1\), \(q>1\) and \(I_\alpha \) is the Riesz potential of order \(\alpha \in (0,N).\) We introduce and study the Coulomb–Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.

Mathematics Subject Classification

35Q55 (35J91, 35J47, 35J50, 31B35) 



C.M. would like to thank Antonio Ambrosetti for drawing his attention to several questions related to nonlinear Schrödinger-Poisson systems. J.V.S was funded by the Fonds de la Recherche Scientifique—FNRS (Grant T.1110.14)


  1. 1.
    Ackermann, N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math Z. 248, 423–443 (2004)Google Scholar
  2. 2.
    Ackermann, N.: A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations. J. Funct. Anal. 234, 277–320 (2006)Google Scholar
  3. 3.
    Adams, D.R., Hedberg, L.I.: Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften. 314, Springer (1996)Google Scholar
  4. 4.
    Ambrosetti, A.: On Schrödinger–Poisson systems. Milan J. Math. 76, 257–274 (2008)Google Scholar
  5. 5.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)Google Scholar
  6. 6.
    Bao, W., Mauser, N.J., Stimming, H.P.: Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger–Poisson-\(X\alpha \) model. Commun. Math. Sci. 1(4), 809–828 (2003)Google Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Bellazzini, J., Ghimenti, M., Ozawa, T.: Sharp lower bounds for Coulomb energy. Math. Res. Lett. 23(3), 621–632 (2016)Google Scholar
  9. 9.
    Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger–Maxwell equations. Topol. Methods Nonlinear Anal. 11(2), 283–293 (1998)Google Scholar
  10. 10.
    Benedek, A., Panzone, R.: The space \(L^P\), with mixed norm. Duke Math. J. 28, 301–324 (1961)Google Scholar
  11. 11.
    Boas, R.P., Jr.: Some uniformly convex spaces. Bull. Am. Math. Soc. 46, 304–311 (1940)Google Scholar
  12. 12.
    Bogachev, V.I.: Measure theory. Springer, Berlin (2007)Google Scholar
  13. 13.
    Bokanowski, O., López, J.L., Soler, J.: On an exchange interaction model for quantum transport: the Schrödinger–Poisson–Slater system. Math. Models Methods Appl. Sci. 13(10), 1397–1412 (2003)Google Scholar
  14. 14.
    Bonheure, D., Mercuri, C.: Embedding theorems and existence results for nonlinear Schrödinger–Poisson systems with unbounded and vanishing potentials. J. Differ. Equ. 251(4–5), 1056–1085 (2011)Google Scholar
  15. 15.
    Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York (2011)Google Scholar
  16. 16.
    Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)Google Scholar
  17. 17.
    Carleson, L.: Selected problems on exceptional sets. Van Nostrand Mathematical Studies, No. 13. Van Nostrand, Princeton, Toronto, London (1967)Google Scholar
  18. 18.
    Catto, I., Dolbeault, J., Sánchez, O., Soler, J.: Existence of steady states for the Maxwell-Schrödinger–Poisson system: exploring the applicability of the concentration-compactness principle. Math. Models Methods Appl. Sci. 23(10), 1915–1938 (2013)Google Scholar
  19. 19.
    D’Aprile, T., Mugnai, D.: Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations. Proc. Roy. Soc. Edinburgh Sect. A. 134(5), 893–906 (2004)Google Scholar
  20. 20.
    Day, M.M.: Some more uniformly convex spaces. Bull. Am. Math. Soc. 47, 504–507 (1941)Google Scholar
  21. 21.
    Del Pino, M., Dolbeault, J.: Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81(9), 847–875 (2002)Google Scholar
  22. 22.
    Di Cosmo, J., Van Schaftingen, J.: Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field. J. Differ. Equ. 259, 596–627 (2015)Google Scholar
  23. 23.
    Duoandikoetxea, J.: Fractional integrals on radial functions with applications to weighted inequalities. Ann. Mat. Pura Appl. 192(4), 553–568 (2013)Google Scholar
  24. 24.
    Gagliardo, E.: Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 7, 102–137 (1958)Google Scholar
  25. 25.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, Berlin (1983)Google Scholar
  26. 26.
    Fröhlich, J., Lieb, E.H., Loss, M.: Stability of Coulomb systems with magnetic fields. I. The one-electron atom. Comm. Math. Phys. 104(2), 251–270 (1986)Google Scholar
  27. 27.
    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
  28. 28.
    Koskela, M.: Some generalizations of Clarkson’s inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., pp. 634–677, pp. 89–93 (1979)Google Scholar
  29. 29.
    Lebedev, N.N.: Special functions and their applications, translated by Silverman RA. Prentice–Hall, Englewood Cliffs (1965)Google Scholar
  30. 30.
    Le Bris, C., Lions, P.L.: From atoms to crystals: a mathematical journey. Bull. Am. Math. Soc. (N.S.). 42(3), 291–363 (2005)Google Scholar
  31. 31.
    Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. (118), 349–374 (1983)Google Scholar
  32. 32.
    Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, RI (2001)Google Scholar
  33. 33.
    Lions, P.L.: Some remarks on Hartree equation. Nonlinear Anal. 5(11), 1245–1256 (1981)Google Scholar
  34. 34.
    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
  35. 35.
    Lions, P.-L.: Solutions of Hartree–Fock equations for Coulomb systems. Comm. Math. Phys. 109(1), 33–97 (1987)Google Scholar
  36. 36.
    Maligranda, L., Sabourova, N.: On Clarkson’s inequality in the real case. Math. Nachr. 280(12), 1363–1375 (2007)Google Scholar
  37. 37.
    Mauser, N.J.: The Schrödinger–Poisson-\(X\alpha \) equation. Appl. Math. Lett. 14(6), 759–763 (2001)Google Scholar
  38. 38.
    Maźya, V.: Sobolev spaces with applications to elliptic partial differential equations, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 342. Springer, Heidelberg (2011)Google Scholar
  39. 39.
    Mercuri, C.: Positive solutions of nonlinear Schrod̈inger–Poisson systems with radial potentials vanishing at infinity. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19(3), 211–227 (2008)Google Scholar
  40. 40.
    Merle, F., Peletier, L.A.: Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth. II. The nonradial case. J. Funct. Anal. 105(1), 1–41 (1992)Google Scholar
  41. 41.
    Milman, D.: On some criteria for the regularity of spaces of type (B). C. R. (Doklady) Acad. Sci. U.R.S.S. 20, 243–246 (1938)Google Scholar
  42. 42.
    Moroz, Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265(2), 153–184 (2013)Google Scholar
  43. 43.
    Ni, W.-M.: A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ. Math. J. 31, 801–807 (1982)Google Scholar
  44. 44.
    Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa. 13(3), 115–162 (1959)Google Scholar
  45. 45.
    Ohtsuka, M.: Capacité d’ensembles de Cantor généralisés. Nagoya Math. J. 11, 151–160 (1957)Google Scholar
  46. 46.
    du Plessis, N.: An introduction to potential theory. University Mathematical Monographs, vol. 7. Oliver and Boyd, Edinburgh (1970)Google Scholar
  47. 47.
    Rubin, B.S.: One-dimensional representation, inversion and certain properties of Riesz potentials of radial functions. Mat. Zametki. 34(4), 521–533 (1983)Google Scholar
  48. 48.
    Ruiz, D.: The Schröinger–Poisson equation under the effect of a nonlinear local term . J. Funct. Anal. 237(2), 655–674 (2006)Google Scholar
  49. 49.
    Ruiz, D.: On the Schrödinger–Poisson–Slater system: behavior of minimizers, radial and nonradial cases. Arch. Ration. Mech. Anal. 198(1), 349–368 (2010)Google Scholar
  50. 50.
    Slater, J.: A Simplification of the Hartree–Fock Method. Phys. Rev. 81, 385–390 (1951)Google Scholar
  51. 51.
    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, vl. 30. Princeton University Press, Princeton (1970)Google Scholar
  52. 52.
    Strauss, W.A.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149–162 (1977)Google Scholar
  53. 53.
    Su, J., Wang, Z.-Q., Willem, M.: Nonlinear Schrödinger equations with unbounded and decaying radial potentials. Commun. Contemp. Math. 9(4), 571–583 (2007)Google Scholar
  54. 54.
    Su, J., Wang, Z.-Q., Willem, M.: Weighted Sobolev embedding with unbounded and decaying radial potentials. J. Differ. Equ. 238(1), 201–219 (2007)Google Scholar
  55. 55.
    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
  56. 56.
    Trudinger, N.S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa. 27(3), 265–308 (1973)Google Scholar
  57. 57.
    Van Schaftingen, J.: Interpolation inequalities between Sobolev and Morrey–Campanato spaces, A common gateway to concentration–compactness and Gagliardo–Nirenberg. Port. Math. 71(3–4), 159–175 (2014)Google Scholar
  58. 58.
    Willem, M.: Minimax theorems. In: Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston Inc., Boston, MA (1996)Google Scholar
  59. 59.
    Willem, M., Functional analysis: fundamentals and applications. Cornerstones, vol. XIV. Birkhäuser, Basel (2013)Google Scholar
  60. 60.
    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)Google Scholar
  61. 61.
    Yosida, K.: Functional analysis, 6th edn. Grundlehren der Mathematischen Wissenschaften, vol. 123. Springer, Berlin, New York (1980)Google Scholar

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Authors and Affiliations

  • Carlo Mercuri
    • 1
  • Vitaly Moroz
    • 1
  • Jean Van Schaftingen
    • 2
  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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