Advertisement

Non-radial sign-changing solutions for the Schrödinger–Poisson problem in the semiclassical limit

  • Isabella Ianni
  • Giusi VairaEmail author
Article

Abstract

We study the following system of equations known as Schrödinger–Poisson problem
$${\left\{\begin{array}{ll}-\epsilon^2\Delta \upsilon + \upsilon +\phi \upsilon=f(\upsilon)&\quad\mbox{in}\mathbb R^N\\-\Delta\phi =a_N \upsilon^2 &\quad \mbox{in} \mathbb R^N\\\phi\rightarrow 0 &\quad\mbox{as} |x|\rightarrow +\infty\end{array}\right.}$$
where \({{{\epsilon > 0}}}\) is a small parameter, \({{{f{:}\;\mathbb{R}\rightarrow\mathbb{R}}}}\) is given, N ≥ 3 , a N is the surface measure of the unit sphere in \({{{\mathbb{R}^{N}}}}\) and the unknowns are \({{{\upsilon, \phi{:}\;\mathbb{R}^{N}\rightarrow\mathbb{R}}}}\) . We construct non-radial sign-changing multi-peak solutions in the semiclassical limit. The peaks are displaced in suitable symmetric configurations and collapse to the same point as \({{{\epsilon}}}\)→ 0. The proof is based on the Lyapunov–Schmidt reduction.

Keywords

Schrödinger–Poisson problem Semiclassical limit Cluster solutions Sign-changing solutions Variational methods Lyapunov–Schmidt reduction 

Mathematics Subject Classification

35B40 35J20 35J61 35Q40 35Q55 

References

  1. 1.
    Ambrosetti A.: On Schrödinger–Poisson systems. Milan J. Math. 76, 257–274 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on \({{{\mathbb{R}^{N}}}}\). Birkhäuser, Boston (2005)Google Scholar
  3. 3.
    Azzollini A., Pomponio A.: Ground state solutions for the nonlinear Schrödinger–Maxwell equations. J. Math. Anal. Appl. 345, 90–108 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benci V., Fortunato D.: An eigenvalue problem for the Schrödinger–Maxwell equations. Top. Methods Nonlinear Anal. 11(2), 283–293 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Benci V., Fortunato D.: Solitary waves of the nonlinear Klein–Gordon equation coupled with the Maxwell equations. Rev. Math. Phys. 14(4), 409–420 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benguria R., Brezis H., Lieb E.H.: The Thomas–Fermi–von Weizscker theory of atoms and molecules. Commun. Math. Phys. 79, 167–180 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Catto I., Lions P.L.: Binding of atoms and stability of molecules in Hartree and Thomas–Fermi type theories. Part 1: a necessary and sufficient condition for the stability of general molecular system. Commun. Partial Differ. Equ. 17, 1051–1110 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D’Aprile T., Mugnai D.: Non-existence results for the coupled Klein–Gordon–Maxwell equations. Adv. Nonlinear Stud. 4, 307–322 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    D’Aprile T., Mugnai D.: Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations. Proc. R. Soc. Edinb. Sect. 134, 893–906 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D’Aprile T., Pistoia A.: On the number of sign-changing solutions of a semiclassical nonlinear Schrödinger equation. Adv. Differ. Equ. 12(7), 737–758 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    D’Aprile T., Wei J.: Standing waves in the Maxwell–Schrödinger equation and an optimal configuration problem. Calc. Var. 25, 105–137 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    D’Aprile T., Wei J.: On bound states concentrating on spheres for the Maxwell–Schrödinger equation. SIAM J. Math. Anal. 37, 321–342 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    del Pino M., Felmer P., Musso M.: Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries. Bull. Lond. Math. Soc. 35, 513–521 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ianni I.: Sign-changing radial solutions for the Schrödinger–Poisson–Slater problem. TMNA 41(2), 365–385 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kikuchi H.: On the existence of solutions for elliptic system related to the Maxwell–Schrödinger equations. Nonlinear Anal. 67, 1445–1456 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lieb E.H.: Thomas–Fermi and related theories and molecules. Rev. Mod. Phys. 53, 603–641 (1981)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lions P.L.: Solutions of Hartree–Fock equations for Coulomb systems. Commun. Math. Phys. 109, 33–97 (1984)CrossRefGoogle Scholar
  19. 19.
    Markowich, P., Ringhofer, C., Schmeiser, C.: Semiconductor Equations. Springer, New York (1990)Google Scholar
  20. 20.
    Ruiz D.: Semiclassical states for coupled Schrödinger–Maxwell equations: concentration around a sphere. M3AS 15, 141–164 (2005)zbMATHGoogle Scholar
  21. 21.
    Ruiz D.: The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Slater J.C.: A simplification of the Hartree–Fock method. Phys. Rev. 81, 385–390 (1951)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaSeconda Università degli Studi di NapoliCasertaItaly
  2. 2.sezione di Matematica, Dipartimento di Scienze di Base e Applicate per l’IngegneriaUniversità La Sapienza di RomaRomeItaly

Personalised recommendations