January  2020, 19(1): 279-292. doi: 10.3934/cpaa.20200015

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

* Corresponding author

Received  December 2018 Revised  March 2019 Published  July 2019

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

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.

Citation: Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015
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.

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} $
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