Advanced Search
Article Contents
Article Contents

Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian

  • * Corresponding author

    * Corresponding author 

This work was supported by National Natural Science Foundation of China (No.11671287, No.11771105) and Guangxi Natural Science Foundation (No.2017GXNSFFA198012)

Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • In this paper we prove the existence and multiplicity of subharmonic solutions for nonlinear equations involving the singular $ \phi $-Laplacian. Such equations are in particular motivated by the one-dimensional mean curvature problems and by the acceleration of a relativistic particle of mass one at rest moving on a straight line. Our approach is based on phase-plane analysis and an application of the Poincaré-Birkhoff twist theorem.

    Mathematics Subject Classification: Primary: 34C25, 34B15; Secondary: 37C25.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The relations between the fundamental period $ T_h $ and "energy" $ h $ with various potentials: (a) Toda potential $ G(x) = k(x+{\mathrm{e}}^{-x}) $ with $ k = 1 $; (b) Sublinear potential $ G(x) = \frac{4}{5}|x|^{5/4} $; (c) Harmonic potential $ G(x) = \frac{1}{2}x^{2} $; (d) Superlinear potential $ G(x) = \frac{2}{5}|x|^{5/2} $

  • [1] N. Atakishiev and R. Mir-Kasimov, Generalized coherent states for rela- tivistic model of a linear oscillator, Theor. Math. Phys., 67 (1986), 362-367.
    [2] C. Bereanu and J. Mawhin, Nonlinear Neumann boundary value problems with $\phi$-Laplacian operators, An. Stiint. Univ. Ovidius Constanta Ser. Mat, 12 (2004), 73-82.
    [3] C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular-Laplacian, J. Differential Equations, 243 (2007), 536-557. doi: 10.1016/j.jde.2007.05.014.
    [4] C. Bereanu and J. Mawhin, Multiple periodic solutions of ordinary differ- ential equations with bounded nonlinearities and $\phi$-Laplacian, NoDEA: Nonlinear Differ. Equ. Appl., 15 (2008), 159-168. doi: 10.1007/s00030-007-7004-x.
    [5] C. Bereanu and J. Mawhin, Boundary value problems for some nonlinear systems with singular $\varphi$-Laplacian, J. Fixed Point Theory Appl., 4 (2008), 57-75. doi: 10.1007/s11784-008-0072-7.
    [6] C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., 140 (2012), 2713-2720. doi: 10.1090/S0002-9939-2011-11101-8.
    [7] A. Boscaggin and G. Feltrin, Postive periodic solutions to an indefinte Minkowski-curvature equation, arXiv: 1805.06659.
    [8] A. Boscaggin and M. Garrione, Sign-changing subharmonic solutions to unforced equations with singular $\phi$-Laplacian, Differential and Difference Equations with Applications, Springer Proceedings in Mathematics and Statistics, 47, 321-329. doi: 10.1007/978-1-4614-7333-6_25.
    [9] T. Ding, Approaches to the Qualitative Theory of Ordinary Differential Equations: Dynamical Systems and Nonlinear Oscilations, Peking University Series in Mathematics, World Scientific Publishing Co. Pte. ltd., Singapore, 2007.
    [10] T. Ding, R. Iannacci and F. Zanolin, On periodic solutions of sublinear Duffing equations, J. Math. Anal. Appl., 158 (1991), 316-332. doi: 10.1016/0022-247X(91)90238-U.
    [11] T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with su- perquadratic potential, J. Differential Equations, 97 (1992), 328-378. doi: 10.1016/0022-0396(92)90076-Y.
    [12] T. Ding and F. Zanolin, Subharmonic solutions of second order nonlinear equations: a time-map approach, Nonlinear Anal., 20 (1993), 509-532. doi: 10.1016/0362-546X(93)90036-R.
    [13] W. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983) 341-346. doi: 10.2307/2044730.
    [14] T. Donde and F. Zanolin, Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach, arXiv: 1901.09406.
    [15] A. Fonda, R. Manásevich and F. Zanolin, Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal., 24 (1993), 1294-1294. doi: 10.1137/0524074.
    [16] A. Fonda and M. Ramos, Large-amplitude subharmonic oscillations for s- calar second-order differential equations with asymmetric nonlinearities, J. Differential Equations, 109 (1994), 354-372. doi: 10.1006/jdeq.1994.1055.
    [17] A. Fonda and A. Sfecci, Periodic solutions of weakly coupled superlinear systems, J. Differential Equations, 260 (2016), 2150-2162. doi: 10.1016/j.jde.2015.09.056.
    [18] A. Fonda and A. Sfecci, A general method for the existence of periodic solutions of differential equations in the plane, J. Differential Equations, 252 (2012), 1369-1391. doi: 10.1016/j.jde.2011.08.005.
    [19] A. Fonda and R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal., 8 (2019), 583-602. doi: 10.1515/anona-2017-0040.
    [20] J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math., 128 (1988), 139-151. doi: 10.2307/1971464.
    [21] J. Ginocchio, Relativistic symmetries in nuclei and hadrons, Phys. Rep., 414 (2005), 165-261. doi: 10.1016/j.physrep.2005.04.003.
    [22] Z. Guo and J. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419-430. doi: 10.1112/S0024610703004563.
    [23] Q. Jiang and C. Tang, Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 328 (2007), 380-389. doi: 10.1016/j.jmaa.2006.05.064.
    [24] J. Kim and H. Lee, Nonlinear resonance and chaos in the relativistic phase space for driven nonlinear systems, Phys. Rev. E, 52 (1995), 473-480.
    [25] J. Kim and H. Lee, Relativistic chaos in the driven harmonic oscillator, Phys. Rev. E, 51 (1995), 1579-1581.
    [26] A. Kolovsky, Relativistic chaos for an electron in a standing microwave field, EPL-Europhysics Lette., 41 (1998), 257.
    [27] D. Kulikov and R. Tutik, Oscillator model for the relativistic fermion-boson system, Phys. Lette. A, 372 (2008), 7105-7108.
    [28] J. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J., 17 (1950), 457-475.
    [29] Z. Opial, Sur les solutions périodiques de léquation différentielle $x''+ g(x) = p(t)$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 8 (1960), 151-156.
    [30] D. Qian, Infinity of Subharmonics for Asymmetric Duffing Equations with the Lazer-Leach-Dancer Condition, J. Differential Equations, 171 (2001), 233-250. doi: 10.1006/jdeq.2000.3847.
    [31] D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725. doi: 10.1137/S003614100343771X.
    [32] D. Qian, P. J. Torres and P. Wang, Periodic solutions of second order equations via rotation numbers, J. Differential Equations, 266 (2019), 4746-4768. doi: 10.1016/j.jde.2018.10.010.
    [33] C. Rebelo, A note on the Poincaré-Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlinear Anal., 29 (1997), 291-311. doi: 10.1016/S0362-546X(96)00065-X.
  • 加载中



Article Metrics

HTML views(414) PDF downloads(276) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint