Sign-Changing Subharmonic Solutions to Unforced Equations with Singular ϕ-Laplacian

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 47)


We prove the existence of infinitely many subharmonic solutions (with a precise nodal characterization) to the equation
$$\displaystyle{\Big( \frac{u^{\prime}} {\sqrt{1 - {u^{\prime}}^{2}}}\Big)^{\prime} + g(t,u) = 0,}$$
in the unforced case g(t,0) ≡ 0. The proof is performed via the Poincaré–Birkhoff fixed point theorem.





The authors wish to thank SISSA for the financial support which has given the pleasant opportunity of taking part in the International Conference on Differential and Difference Equations and Applications in Ponta Delgada, July 2011.


  1. 1.
    Bereanu, C., Mawhin, J.: Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian. J. Differ. Equ. 243, 536–557 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bereanu, C., Jebelean, P., Mawhin, J.: Periodic solutions of pendulum-like perturbations of singular and bounded ϕ-Laplacians. J. Dynam. Differ. Equ. 22, 463–471 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bereanu, C., Jebelean, P., Mawhin, J.: Variational methods for nonlinear perturbations of singular ϕ-Laplacians. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 22(9), 89–111 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bereanu, C., Torres, P.: Existence of at least two periodic solutions of the forced relativistic pendulum. Proc. Amer. Math. Soc. 140, 2713–2719 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bonheure, D., Habets, P., Obersnel, F., Omari, P.: Classical and non-classical solutions of a prescribed curvature equation. J. Differ. Equ. 243, 208–237 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Boscaggin, A.: Subharmonic solutions of planar Hamiltonian systems: a rotation number approach. Adv. Nonlinear Stud. 11, 77–103 (2011)MathSciNetMATHGoogle Scholar
  7. 7.
    Boscaggin, A., Zanolin, F.: Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete Contin. Dyn. Syst. 33, 89–110 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ding, W.-Y.: Fixed points of twist mappings and periodic solutions of ordinary differential equations (Chinese). Acta Math. Sinica 25, 227–235 (1982)MathSciNetMATHGoogle Scholar
  9. 9.
    Ding, T., Iannacci, R., Zanolin, F.: Existence and multiplicity results for periodic solutions of semilinear Duffing equations. J. Differ. Equ. 105, 364–409 (1993)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fonda, A., Toader, R.: Periodic solutions of pendulum-like Hamiltonian systems in the plane Adv. Nonlinear Stud. 12, 395–408 (2012)MathSciNetMATHGoogle Scholar
  11. 11.
    Marò, S.: Periodic solutions of a forced relativistic pendulum via twist dynamics to appear on Topol. Methods Nonlinear Anal.Google Scholar
  12. 12.
    Mawhin, J.: Stability and bifurcation theory for non-autonomous differential equations (Cetraro, 2011), Lecture Notes in Math. 2065, to appear on Topol. Methods Nonlinear Anal. Springer, Berlin, (2013), 103–184Google Scholar
  13. 13.
    Obersnel, F. Omari, P.: Multiple bounded variation solutions of a periodically perturbed sine-curvature equation. Commun. Contemp. Math. 13, 863–883 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rebelo, C., Zanolin, F.: Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities. Trans. Amer. Math. Soc. 348, 2349–2389 (1996)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Zanini, C.: Rotation numbers, eigenvalues, and the Poincaré-Birkhoff theorem. J. Math. Anal. Appl. 279, 290–307 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TorinoTorinoItaly
  2. 2.Dipartimento di Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

Personalised recommendations