Multiple solutions to a magnetic nonlinear Choquard equation

  • Silvia Cingolani
  • Mónica Clapp
  • Simone Secchi


We consider the stationary nonlinear magnetic Choquard equation
$$(- {\rm i}\nabla+ A(x))^{2}u + V (x)u = \left(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\right) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}$$
where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, \({\alpha \in (0, N)}\) and 2 − (α/N) < p < (2Nα)/(N−2). We assume that both A and V are compatible with the action of some group G of linear isometries of \({\mathbb{R}^{N}}\) . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition
$$u(gx) = \tau(g)u(x)\quad{\rm for\, all }\ g \in G,\;x \in \mathbb{R}^{N},$$
where \({\tau : G \rightarrow \mathbb{S}^{1}}\) is a given group homomorphism into the unit complex numbers.

Mathematics Subject Classification (2010)

35Q55 35Q40 35J20 35B06 


Nonlinear Choquard equation Nonlocal nonlinearity Electromagnetic potential Multiple solutions Intertwining solutions 


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  1. 1.
    Ackermann N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cingolani S., Clapp M.: Intertwining semiclassical bound states to a nonlinear magnetic equation. Nonlinearity 22, 2309–2331 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cingolani, S., Clapp, M., Secchi, S.: Intertwining semiclassical solutions to a Schrödinger-Newton system, preprintGoogle Scholar
  5. 5.
    Cingolani S., Secchi S., Squassina M.: Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc. Roy. Soc. Edinburgh 140 A, 973–1009 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Clapp M., Puppe D.: Critical point theory with symmetries. J. Reine Angew. Math. 418, 1–29 (1991)MathSciNetzbMATHGoogle Scholar
  7. 7.
    tom Dieck T.: Transformation Groups. Walter de Gruyter, Berlin-New York (1987)CrossRefzbMATHGoogle Scholar
  8. 8.
    Esteban M.J., Lions P.L.: Stationary solutions of nonlinear Schrödinger equations with an external magnetic field. In: (eds) Partial Differential Equations and the Calculus of Variations, Vol. 1, pp. 401–449. Progr. Nonlinear Differential Equations Appl., 1, Birkhäuser Boston, Boston, MA (1989)Google Scholar
  9. 9.
    Fadell E., Husseini S., Rabinowitz P.H.: Borsuk-Ulam theorems for \({\mathbb{S}^{1} }\) -actions and applications. Trans. Am. Math. Soc. 274, 345–359 (1982)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fröhlich, J., Lenzmann, E.: Mean-field limit of quantum Bose gases and nonlinear Hartree equation. In: Séminaire: Équations aux Dérivées Partielles 2003–2004, Exp. No. XIX, pp. 26. École Polytech., PalaiseauGoogle Scholar
  11. 11.
    Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}^{N} }\) , Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7A. pp. 369–402. Academic Press, New York–LondonGoogle Scholar
  12. 12.
    Ginibre J., Velo G.: On a class of nonlinear Schrödinger equations with nonlocal interaction. Math. Z. 170, 109–136 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lieb E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977)MathSciNetGoogle Scholar
  14. 14.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Math. vol. 14. American Mathematical Society, Providence RI (1997)Google Scholar
  15. 15.
    Lieb E.H., Simon B.: The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys. 53, 185–194 (1977)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lions P.-L.: The Choquard equation and related questions. Nonlinear Anal. TMA 4, 1063–1073 (1980)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lions, P.-L.: The concentration-compacteness principle in the calculus of variations. The locally compact case. Ann. Inst. Henry Poincaré, Analyse Non Linéaire vol. 1. pp. 109–145 and 223–283 (1984)Google Scholar
  18. 18.
    Lions, P.-L.: Symmetries and the concentration-compacteness method. In: Nonlinear Variational Problems, pp. 47–56. Pitman, London, (1985)Google Scholar
  19. 19.
    Ma L., Zhao L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Moroz, I.M., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger-Newton equations. Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity vol. 15. pp. 2733–2742 (1998)Google Scholar
  21. 21.
    Moroz I.M., Tod P.: An analytical approach to the Schrödinger-Newton equations. Nonlinearity 12, 201–216 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nolasco M.: Breathing modes for the Schrödinger–Poisson system with a multiple–well external potential. Commun. Pure Appl. Anal. 9, 1411–1419 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Palais R.: The principle of symmetric criticallity. Comm. Math. Phys. 69, 19–30 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Penrose R.: On gravity’s role in quantum state reduction. Gen. Rel. Grav. 28, 581–600 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Penrose R.: Quantum computation, entanglement and state reduction. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356, 1927–1939 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Penrose R.: The Road to Reality. A Complete Guide to the Laws of the Universe. Alfred A. Knopf Inc., New York (2005)zbMATHGoogle Scholar
  27. 27.
    Struwe M.: Variational Methods. Springer, Berlin-Heidelberg (1996)zbMATHGoogle Scholar
  28. 28.
    Secchi S.: A note on Schrödinger–Newton systems with decaying electric potential. Nonlinear Anal. 72, 3842–3856 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tod P.: The ground state energy of the Schrödinger-Newton equation. Phys. Lett. A 280, 173–176 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wei J., Winter M.: Strongly interacting bumps for the Schrödinger–Newton equation. J. Math. Phys. 50, 012905 (2009)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Willem M.: Minimax theorems. PNLDE vol. 24. Birkhäuser, Boston-Basel-Berlin (1996)CrossRefGoogle Scholar
  32. 32.
    Zhang Z., Küpper T., Hu A., Xia H.: Existence of a nontrivial solution for Choquard’s equation. Acta Math. Sci. Ser. B Engl. Ed. 26, 460–468 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Silvia Cingolani
    • 1
  • Mónica Clapp
    • 2
  • Simone Secchi
    • 3
  1. 1.Dipartimento di MatematicaPolitecnico di BariBariItaly
  2. 2.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMéxico D.F.Mexico
  3. 3.Dipartimento di Matematica ed ApplicazioniUniversità di Milano-BicoccaMilanoItaly

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