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

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Abstract

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.

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References

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

  2. Ackermann, N.: A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations. J. Funct. Anal. 234, 277–320 (2006)

  3. Adams, D.R., Hedberg, L.I.: Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften. 314, Springer (1996)

  4. Ambrosetti, A.: On Schrödinger–Poisson systems. Milan J. Math. 76, 257–274 (2008)

  5. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

  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)

  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)

  8. Bellazzini, J., Ghimenti, M., Ozawa, T.: Sharp lower bounds for Coulomb energy. Math. Res. Lett. 23(3), 621–632 (2016)

  9. Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger–Maxwell equations. Topol. Methods Nonlinear Anal. 11(2), 283–293 (1998)

  10. Benedek, A., Panzone, R.: The space \(L^P\), with mixed norm. Duke Math. J. 28, 301–324 (1961)

  11. Boas, R.P., Jr.: Some uniformly convex spaces. Bull. Am. Math. Soc. 46, 304–311 (1940)

  12. Bogachev, V.I.: Measure theory. Springer, Berlin (2007)

  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)

  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)

  15. Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York (2011)

  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)

  17. Carleson, L.: Selected problems on exceptional sets. Van Nostrand Mathematical Studies, No. 13. Van Nostrand, Princeton, Toronto, London (1967)

  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)

  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)

  20. Day, M.M.: Some more uniformly convex spaces. Bull. Am. Math. Soc. 47, 504–507 (1941)

  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)

  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)

  23. Duoandikoetxea, J.: Fractional integrals on radial functions with applications to weighted inequalities. Ann. Mat. Pura Appl. 192(4), 553–568 (2013)

  24. Gagliardo, E.: Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 7, 102–137 (1958)

  25. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, Berlin (1983)

  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)

  27. Ianni, I., Ruiz, D.: Ground and bound states for a static Schrödinger–Poisson–Slater problem. Commun. Contemp. Math. 14(1), 1250003, 22 (2012)

  28. Koskela, M.: Some generalizations of Clarkson’s inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., pp. 634–677, pp. 89–93 (1979)

  29. Lebedev, N.N.: Special functions and their applications, translated by Silverman RA. Prentice–Hall, Englewood Cliffs (1965)

  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)

  31. Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. (118), 349–374 (1983)

  32. Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, RI (2001)

  33. Lions, P.L.: Some remarks on Hartree equation. Nonlinear Anal. 5(11), 1245–1256 (1981)

  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)

  35. Lions, P.-L.: Solutions of Hartree–Fock equations for Coulomb systems. Comm. Math. Phys. 109(1), 33–97 (1987)

  36. Maligranda, L., Sabourova, N.: On Clarkson’s inequality in the real case. Math. Nachr. 280(12), 1363–1375 (2007)

  37. Mauser, N.J.: The Schrödinger–Poisson-\(X\alpha \) equation. Appl. Math. Lett. 14(6), 759–763 (2001)

  38. Maźya, V.: Sobolev spaces with applications to elliptic partial differential equations, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 342. Springer, Heidelberg (2011)

  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)

  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)

  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)

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

  43. Ni, W.-M.: A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ. Math. J. 31, 801–807 (1982)

  44. Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa. 13(3), 115–162 (1959)

  45. Ohtsuka, M.: Capacité d’ensembles de Cantor généralisés. Nagoya Math. J. 11, 151–160 (1957)

  46. du Plessis, N.: An introduction to potential theory. University Mathematical Monographs, vol. 7. Oliver and Boyd, Edinburgh (1970)

  47. Rubin, B.S.: One-dimensional representation, inversion and certain properties of Riesz potentials of radial functions. Mat. Zametki. 34(4), 521–533 (1983)

  48. Ruiz, D.: The Schröinger–Poisson equation under the effect of a nonlinear local term . J. Funct. Anal. 237(2), 655–674 (2006)

  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)

  50. Slater, J.: A Simplification of the Hartree–Fock Method. Phys. Rev. 81, 385–390 (1951)

  51. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, vl. 30. Princeton University Press, Princeton (1970)

  52. Strauss, W.A.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149–162 (1977)

  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)

  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)

  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)

  56. Trudinger, N.S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa. 27(3), 265–308 (1973)

  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)

  58. Willem, M.: Minimax theorems. In: Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston Inc., Boston, MA (1996)

  59. Willem, M., Functional analysis: fundamentals and applications. Cornerstones, vol. XIV. Birkhäuser, Basel (2013)

  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)

  61. Yosida, K.: Functional analysis, 6th edn. Grundlehren der Mathematischen Wissenschaften, vol. 123. Springer, Berlin, New York (1980)

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Acknowledgments

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)

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Authors and Affiliations

  1. Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, Wales, UK

    Carlo Mercuri & Vitaly Moroz

  2. Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2 bte L7.01.01, 1348, Louvain-la-Neuve, Belgium

    Jean Van Schaftingen

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  1. Carlo Mercuri
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Correspondence to Vitaly Moroz.

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Communicated by A. Malchiodi.

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Mercuri, C., Moroz, V. & Van Schaftingen, J. Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency. Calc. Var. 55, 146 (2016). https://doi.org/10.1007/s00526-016-1079-3

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