Advertisement

Klein–Gordon–Maxwell–Proca type systems in the electro-magneto-static case: the high dimensional case

  • Emmanuel HebeyEmail author
  • Pierre-Damien Thizy
Article
  • 75 Downloads

Abstract

We investigate Klein–Gordon–Maxwell–Proca type systems in the context of closed n-dimensional manifolds with \(n \ge 4\). We prove existence of solutions and compactness of the system both in the subcritical and in the critical case.

Mathematics Subject Classification

35J47 58J05 

Notes

Acknowledgements

The authors would like to express their sincere gratitude to the referee for his/her useful comments. The second author is supported by the ANR-BLADE-JC.

References

  1. 1.
    Benci, V., Fortunato, D.: Solitary waves of the nonlinear Klein–Gordon field equation coupled with the Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benci, V., Fortunato, D.: Spinning \(Q\)-balls for the Klein–Gordon–Maxwell equations. Commun. Math. Phys. 295, 639–668 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Biquard, O.: Polycopié on differential geometry and global analysis, Unpublished Notes (2008)Google Scholar
  4. 4.
    Brézis, H., Li, Y.Y.: Some nonlinear elliptic equations have only constant solutions. J. Partial Differ. Equ. 19, 208–217 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Clapp, M., Ghimenti, M., Micheletti, A.M.: Semiclassical states for a static supercritical Klein–Gordon–Maxwell–Proca system on a closed Riemannian manifold. Commun. Contemp. Math. 18, 1550039 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    d’Avenia, P., Medreski, J., Pomponio, P.: Vortex ground states for Klein–Gordon–Maxwell–Proca type systems. J. Math. Phys. 58, 041503 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Druet, O.: From one bubble to several bubbles: the low-dimensional case. J. Differ. Geom. 63, 399–473 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Druet, O.: Compactness for Yamabe metrics in low dimensions. Int. Math. Res. Not. 23, 1143–1191 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Druet, O., Hebey, E.: Existence and a priori bounds for electrostatic Klein–Gordon–Maxwell systems in fully inhomogeneous spaces. Commun. Contemp. Math. 12, 831–869 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Druet, O., Hebey, E., Vétois, J.: Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian. J. Funct. Anal. 258, 999–1059 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Druet, O., Hebey, E., Vétois, J.: Static Klein–Gordon–Maxwell–Proca systems in \(4\)-dimensional closed manifolds. II. J. Reine Angew. Math 713, 149–179 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Epstein, H., Moschella, U.: de Sitter tachyons and related topics. Commun. Math. Phys. 336(1), 381–430 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equ. 6, 883–901 (1981)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gilbarg, G., Trudinger, N.S.: Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften, vol. 224, 2nd edn. Springer, Berlin (1983)Google Scholar
  16. 16.
    Goldhaber, A.S., Nieto, M.M.: Terrestrial and extraterrestrial limits on the photon mass. Rev. Mod. Phys. 43, 277–296 (1971)CrossRefGoogle Scholar
  17. 17.
    Goldhaber, A.S., Nieto, M.M.: Photon and Graviton mass limits. Rev. Mod. Phys. 82, 939–979 (2010)CrossRefGoogle Scholar
  18. 18.
    Hebey, E.: Solitary waves in critical Abelian Gauge theories. Discret. Contin. Dyn. Syst. 32(5), 1747–1761 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hebey, E.: Compactness and stability for nonlinear elliptic equations. Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zurich (2014)Google Scholar
  20. 20.
    Hebey, E., Thizy, P.D.: Stationary Kirchhoff systems in closed high dimensional manifolds. Commun. Contemp. Math 18(2), 1550028 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hebey, E.: Klein–Gordon–Maxwell–Proca type systems in the electro-magneto-static case. J. Partial Differ. Equ. 31, 119–58 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hebey, E., Truong, T.T.: Static Klein–Gordon–Maxwell–Proca systems in \(4\)-dimensional closed manifolds. J. Reine Angew. Math. 667, 221–248 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hebey, E., Wei, J.: Resonant states for the static Klein–Gordon–Maxwell–Proca system. Math. Res. Lett. 19, 953–967 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li, Y.Y., Zhu, M.: Yamabe type equations on three dimensional Riemannian manifolds. Commun. Contemp. Math. 1, 1–50 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lin, C.S., Ni, W.M., Takagi, I.: Large amplitude stationary solutions to a chemotaxis system. J. Differ. Equ. 72(1), 127 (1988)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Luo, J., Gillies, G.T., Tu, L.C.: The mass of the photon. Rep. Prog. Phys. 68, 77–130 (2005)CrossRefGoogle Scholar
  27. 27.
    Marques, F.C.: A priori estimates for the Yamabe problem in the non-locally conformally flat case. J. Differ. Geom. 71, 315–346 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Robert, F.: Existence et asymptotiques optimales des fonctions de Green des opérateurs elliptiques d’ordre deux, Unpublished Notes (2009)Google Scholar
  29. 29.
    Schoen, R.M.: Lecture notes from courses at Stanford, written by D. Pollack, preprint (1988)Google Scholar
  30. 30.
    Thizy, P.D.: Klein–Gordon–Maxwell equations in high dimensions. Commun. Pure Appl. Anal. 14, 1097–1125 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Cergy-PontoiseCergy-Pontoise CedexFrance
  2. 2.Institut Camille JordanUniversité Claude Bernard Lyon 1Villeurbanne CedexFrance

Personalised recommendations