Skip to main content
SpringerLink
Log in

On the structure of the Nehari set associated to a Schrödinger-Poisson system with prescribed mass: old and new results

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we apply the fibering method of Pohozaev and the notion of extremal values introduced by Il’yasov to a Schrödinger—Poisson system, with prescribed L2 norm of the unknown, in the whole ℝ3. The method makes clear the role played by the special exponents p = 3, p = 8/3, p = 10/3.

In addition to showing that old results can be obtained in a unified way, we exhibit also new ones.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Bellazzini, L. Jeanjean and T. Luo, Existence and instability of standing waves with prescribed norm for a class of Schrödinger—Poisson equations, Proceedings of the London Mathematical Society 107 (2013), 303–339.

    Article  MathSciNet  Google Scholar 

  2. J. Bellazzini and G. Siciliano, Stable standing waves for a class of nonlinear Schrödinger—Poisson equations, Zeitschrift für Angewandte Mathematik und Physik 62 (2010), 267–280.

    Article  Google Scholar 

  3. J. Bellazzini and G. Siciliano, Scaling properties of functionals and existence of constrained minimizers, Journal of Functional Analysis 261 (2011), 2486–2507.

    Article  MathSciNet  Google Scholar 

  4. V. Benci, Hylomorphic solitons, Milan Journal of Mathematics 77 (2009), 271–332.

    Article  MathSciNet  Google Scholar 

  5. V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger—Maxwell equations, Topological Methods in Nonlinear Analysis 11 (1998), 283–293.

    Article  MathSciNet  Google Scholar 

  6. O. Bokanowski and N. Mauser, Local approximation for the Hartree-Fock exchange potential: a deformation approach, Mathematical Models and Methods in Applied Sciences 9 (1999), 941–961.

    Article  MathSciNet  Google Scholar 

  7. I. Catto, J. Dolbeault, O. Sánchez and J. Soler, Existence of steady states for the Maxwell-Schrödinger-Poisson system: exploring the applicability of the concentration-compactness principle, Mathematical Models and Methods in Applied Sciences 23 (2013), 1915–1938.

    Article  MathSciNet  Google Scholar 

  8. I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. I. A necessary and sufficient condition for the stability of general molecular systems, Communications in Partial Differential Equations 17 (1992), 1051–1110.

    Article  MathSciNet  Google Scholar 

  9. M. Colin and T. Watanabe, Standing waves for the nonlinear Schrödinger equation coupled with the Maxwell equation, Nonlinearity 30 (2017), 1920–1947.

    Article  MathSciNet  Google Scholar 

  10. Y. Il’yasov On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient, Topological Methods in Nonlinear Analysis 49 (2017), 683–714.

    MathSciNet  Google Scholar 

  11. Y. Il’yasov and K. Silva, On branches of positive solutions for p-Laplacian problems at the extreme value of the Nehari manifold method, Proceedings of the American Mathematical Society 146 (2018), 2925–2935.

    Article  MathSciNet  Google Scholar 

  12. L. Jeanjean and T. Luo, Sharp nonexistence results of prescribed L2-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Zeitschrift für Angewandte Mathematik und Physik 64 (2013), 937–954.

    Article  MathSciNet  Google Scholar 

  13. L. Jeanjean, T. Luo and Z.-Q. Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, Journal of Differential Equations 259 (2015), 3894–3928.

    Article  MathSciNet  Google Scholar 

  14. E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001

    Google Scholar 

  15. E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Communications in Mathematical Physics 53, (1977) 185–194.

    Article  MathSciNet  Google Scholar 

  16. P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. I, II, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 (1984), 109–145, 223–283.

    Article  MathSciNet  Google Scholar 

  17. P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Communications in Mathematical Physics 109 (1987), 33–97.

    Article  MathSciNet  Google Scholar 

  18. N. J. Mauser, The Schrödinger-Poisson-Xctequation, Applied Mathematics Letters 14 (2001) 759–763.

    Article  MathSciNet  Google Scholar 

  19. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989.

    Google Scholar 

  20. S. I. Pohozaev, The fibration method for solving nonlinear boundary value problems, Trudy Matematicheskogo Instituta imeni V. A. Steklova 192 (1990), 146–163. English translation: Differential Equations and Function Spaces, Proceedings of the Steklov Institute of Mathematics, Vol. 192, American Mathematical Society, Providence, RI, 1992, pp. 157–163.

    MathSciNet  Google Scholar 

  21. D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, Journal of Funcional Analysis 237 (2006), 655–674.

    Article  Google Scholar 

  22. O. Sánchez and J. Soler, Long-time dynamics of the Schrödinger-Poisson-Slater system, Journal of Statistical Physics 114 (2004), 179–204.

    Article  MathSciNet  Google Scholar 

  23. G. Siciliano and K. Silva, The fibering method approach for a non-linear Schrödinger equation coupled with the electromagnetic field, Publicacions Matemàtiques 64 (2020), 373–390.

    Article  MathSciNet  Google Scholar 

  24. K. Silva, On an abstract bifurcation result concerning homogeneous potential operators with applications to PDEs, Journal of Differential Equations 269 (2020), 7643–7675.

    Article  MathSciNet  Google Scholar 

  25. J. C. Slater, A simplification of the Hartree-Fock method, Physics Review 81 (1951) 385–390.

    Article  Google Scholar 

  26. L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations, Journal of Mathematical Analysis and Applications 346 (2008), 155–169.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgment

The authors would like to thank the anonimous referee for the attention paid to the paper and the useful suggestions which improved the reading of the manuscript.

Author information

Authors and Affiliations

  1. Departamento de Matemática - Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090, São Paulo, SP, Brazil

    Gaetano Siciliano

  2. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Rua Samambaia, 74001-970, Goiânia, GO, Brazil

    Kaye Silva

Authors
  1. Gaetano Siciliano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kaye Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gaetano Siciliano.

Additional information

Gaetano Siciliano was partially supported by Fapesp grant 2019/27491-0, CNPq grant 304660/2018-3 and Capes.

Kaye Silva was partially supported by CNPq/Brazil under Grant 408604/2018-2.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Siciliano, G., Silva, K. On the structure of the Nehari set associated to a Schrödinger-Poisson system with prescribed mass: old and new results. Isr. J. Math. 258, 403–451 (2023). https://doi.org/10.1007/s11856-023-2477-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-023-2477-9

Search

Navigation