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Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian
1. | School of Mathematical Sciences, Soochow University, Suzhou, 215006, China |
2. | School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, 541003, China |
3. | School of Mathematical Sciences, Soochow University, Suzhou, 215006, China |
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.
References:
[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. |
show all references
References:
[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. |

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