American Institute of Mathematical Sciences

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May  2021, 20(5): 2117-2138. doi: 10.3934/cpaa.2021060

Periodic solutions of p-Laplacian equations via rotation numbers

Shuang Wang 1,, and  Dingbian Qian 2,

1. 

School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224051, China
2. 

School of Mathematical Sciences, Soochow University, Suzhou 215006, China

* Corresponding author

Received  March 2020 Revised  February 2021 Published  April 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (No. 11901507, No. 12071327 and No. 11671287), the Natural Science Foundation of Jiangsu Province (No. BK20181058), and the Qing Lan Project of the Jiangsu Higher Education Institutions of China

We investigate the existence and multiplicity of periodic solutions of the $ p $-Laplacian equation $ \left(\phi_p(x')\right)'+f(t, x) = 0 $. Both asymptotically linear and partially superlinear nonlinearities are studied, in absence of global existence and uniqueness conditions on the solutions of the associated Cauchy problems and the sign assumption on $ f $. We use a approach of rotation number in the $ p $-polar coordinates transformation, together with the phase-plane analysis of the rotational properties of large solutions and a recent version of Poincaré-Birkhoff theorem for Hamiltonian systems, for obtaining multiplicity results of $ p $-Laplacian equation in terms of the gap between the rotation numbers of referred piecewise $ p $-linear systems at zero and infinity.

Keywords: Periodic solution, $ p $-Laplacian equation, spiral property, rotation number, Poincaré-Birkhoff theorem.
Mathematics Subject Classification: Primary: 34C25, 34B15; Secondary: 34D15.
Citation: Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060
References:
[1]

L. Boccardo, P. Dràbek, D. Giachetti and M. Kuček, Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear Anal., 10 (1986), 1083-1103.  doi: 10.1016/0362-546X(86)90091-X.  Google Scholar

[2]

Z. Cheng and J. Ren, Existence of multiplicity harmonic and subharmonic solutions for second-order quasilinear equation via Poincaré-Birkhoff twist theorem, Math. Methods Appl. Sci., 40 (2017), 6801-6822.  doi: 10.1002/mma.4494.  Google Scholar

[3]

M. Cuesta and J. Gossez, A variational approach to nonresonance with respect to the Fučik spectrum, Nonlinear Anal., 19 (1992), 487-500.  doi: 10.1016/0362-546X(92)90087-U.  Google Scholar

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F. Dalbono and F. Zanolin, Multiplicity results for asymptotically linear equations, using the rotation number approach, Mediterr. J. Math., 4 (2007), 127-149.  doi: 10.1007/s00009-007-0108-z.  Google Scholar

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W. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983), 341-346.  doi: 10.2307/2044730.  Google Scholar

[6]

T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differ. Equ., 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar

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A. Fonda and A. Sfecci, Periodic solutions of weakly coupled superlinear systems, J. Differ. Equ., 260 (2016), 2150-2162.  doi: 10.1016/j.jde.2015.09.056.  Google Scholar

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A. Fonda and A. J. Ureña, A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 679-698.  doi: 10.1016/j.anihpc.2016.04.002.  Google Scholar

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J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math., 128 (1988), 139-151.  doi: 10.2307/1971464.  Google Scholar

[10]

M. García-Huidobro, R. Manásevich and F. Zanolin, A Fredholm-like result for strongly nonlinear second order ODE's, J. Differ. Equ., 114 (1994), 132-167.  doi: 10.1006/jdeq.1994.1144.  Google Scholar

[11]

P. Hartman, On boundary value problems for superlinear second order differential equations, J. Differ. Equ., 26 (1977), 37-53.  doi: 10.1016/0022-0396(77)90097-3.  Google Scholar

[12]

H. Jacobowitz, Periodic solutions of $x''+f(t, x) = 0$ via the Poincaré-Birkhoff theorem, J. Differ. Equ., 20 (1976), 37-52.  doi: 10.1016/0022-0396(76)90094-2.  Google Scholar

[13] V. Lakshmikantham and S. Leela, Differential and integral inequalities; theory and applications, Academic Press, New York and London, 1969.   Google Scholar
[14]

R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with $p$-Laplacian like operators, J. Differ. Equ., 145 (1998), 367-393.  doi: 10.1006/jdeq.1998.3425.  Google Scholar

[15]

R. Manásevich and F. Zanolin, Time-mappings and multiplicity of solutions for the one-dimensional $p$-Laplacian, Nonlinear Anal., 21 (1993), 269-291.  doi: 10.1016/0362-546X(93)90020-S.  Google Scholar

[16]

A. Margheri, C. Rebelo and P. J. Torres, On the use of Morse index and rotation numbers for multiplicity results of resonant BVPs, J. Math. Anal. Appl., 413 (2014), 660-667.  doi: 10.1016/j.jmaa.2013.12.005.  Google Scholar

[17]

X. Ming, S. Wu and J. Liu, Periodic solutions for the 1-dimensional $p$-Laplacian equation, J. Math. Anal. Appl., 325 (2007), 879-888. doi: 10.1016/j.jmaa.2006.02.027.  Google Scholar

[18]

M. del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')+f(t, u) = 0, u(0) = u(T) = 0, p>1$, J. Differ. Equ., 80 (1989), 1-13.  doi: 10.1016/0022-0396(89)90093-4.  Google Scholar

[19]

M. del Pino, R. Manásevich and A. E. Murúa, Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE, Nonlinear Anal., 18 (1992), 79-92.  doi: 10.1016/0362-546X(92)90048-J.  Google Scholar

[20]

M. del Pino and R. Manásevich, Infinitely many $2\pi$-periodic solutions for a problem arising in nonlinear elasticity, J. Differ. Equ., 103 (1993), 260-277.  doi: 10.1006/jdeq.1993.1050.  Google Scholar

[21]

D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach-Dancer condition, J. Differ. Equ., 171 (2001), 233-250.  doi: 10.1006/jdeq.2000.3847.  Google Scholar

[22]

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.  Google Scholar

[23]

D. Qian, L. Chen and X. Sun, Periodic solutions of superlinear impulsive differential equations: a geometric approach, J. Differ. Equ., 258 (2015), 3088-3106.  doi: 10.1016/j.jde.2015.01.003.  Google Scholar

[24]

D. Qian, P. J. Torres and P. Wang, Periodic solutions of second Order equations via rotation Numbers, J. Differ. Equ., 266 (2019), 4746–4768. doi: 10.1016/j.jde.2018.10.010.  Google Scholar

[25]

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.  Google Scholar

[26]

H. Royden, P. Fitzpatrick, Real Analysis, 4$^{th}$ edition, Printice-Hall Inc, Boston, 2010. Google Scholar

[27]

P. Yan and M. Zhang, Rotation number, periodic Fucik spectrum and multiple periodic solutions, Commun. Contemp. Math., 12 (2010), 437-455.  doi: 10.1142/S0219199710003877.  Google Scholar

[28]

M. Zhang, Nonuniform nonresonance at the first eigenvalue of the p-Laplacian, Nonlinear Anal., 29 (1997), 41-51.  doi: 10.1016/S0362-546X(96)00037-5.  Google Scholar

[29]

M. Zhang, Nonuniform nonresonance of semilinear differential equations, J. Differ. Equ., 166 (2000), 33-50.  doi: 10.1006/jdeq.2000.3798.  Google Scholar

show all references

References:
[1]

L. Boccardo, P. Dràbek, D. Giachetti and M. Kuček, Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear Anal., 10 (1986), 1083-1103.  doi: 10.1016/0362-546X(86)90091-X.  Google Scholar

[2]

Z. Cheng and J. Ren, Existence of multiplicity harmonic and subharmonic solutions for second-order quasilinear equation via Poincaré-Birkhoff twist theorem, Math. Methods Appl. Sci., 40 (2017), 6801-6822.  doi: 10.1002/mma.4494.  Google Scholar

[3]

M. Cuesta and J. Gossez, A variational approach to nonresonance with respect to the Fučik spectrum, Nonlinear Anal., 19 (1992), 487-500.  doi: 10.1016/0362-546X(92)90087-U.  Google Scholar

[4]

F. Dalbono and F. Zanolin, Multiplicity results for asymptotically linear equations, using the rotation number approach, Mediterr. J. Math., 4 (2007), 127-149.  doi: 10.1007/s00009-007-0108-z.  Google Scholar

[5]

W. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983), 341-346.  doi: 10.2307/2044730.  Google Scholar

[6]

T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differ. Equ., 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar

[7]

A. Fonda and A. Sfecci, Periodic solutions of weakly coupled superlinear systems, J. Differ. Equ., 260 (2016), 2150-2162.  doi: 10.1016/j.jde.2015.09.056.  Google Scholar

[8]

A. Fonda and A. J. Ureña, A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 679-698.  doi: 10.1016/j.anihpc.2016.04.002.  Google Scholar

[9]

J. Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. Math., 128 (1988), 139-151.  doi: 10.2307/1971464.  Google Scholar

[10]

M. García-Huidobro, R. Manásevich and F. Zanolin, A Fredholm-like result for strongly nonlinear second order ODE's, J. Differ. Equ., 114 (1994), 132-167.  doi: 10.1006/jdeq.1994.1144.  Google Scholar

[11]

P. Hartman, On boundary value problems for superlinear second order differential equations, J. Differ. Equ., 26 (1977), 37-53.  doi: 10.1016/0022-0396(77)90097-3.  Google Scholar

[12]

H. Jacobowitz, Periodic solutions of $x''+f(t, x) = 0$ via the Poincaré-Birkhoff theorem, J. Differ. Equ., 20 (1976), 37-52.  doi: 10.1016/0022-0396(76)90094-2.  Google Scholar

[13] V. Lakshmikantham and S. Leela, Differential and integral inequalities; theory and applications, Academic Press, New York and London, 1969.   Google Scholar
[14]

R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with $p$-Laplacian like operators, J. Differ. Equ., 145 (1998), 367-393.  doi: 10.1006/jdeq.1998.3425.  Google Scholar

[15]

R. Manásevich and F. Zanolin, Time-mappings and multiplicity of solutions for the one-dimensional $p$-Laplacian, Nonlinear Anal., 21 (1993), 269-291.  doi: 10.1016/0362-546X(93)90020-S.  Google Scholar

[16]

A. Margheri, C. Rebelo and P. J. Torres, On the use of Morse index and rotation numbers for multiplicity results of resonant BVPs, J. Math. Anal. Appl., 413 (2014), 660-667.  doi: 10.1016/j.jmaa.2013.12.005.  Google Scholar

[17]

X. Ming, S. Wu and J. Liu, Periodic solutions for the 1-dimensional $p$-Laplacian equation, J. Math. Anal. Appl., 325 (2007), 879-888. doi: 10.1016/j.jmaa.2006.02.027.  Google Scholar

[18]

M. del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')+f(t, u) = 0, u(0) = u(T) = 0, p>1$, J. Differ. Equ., 80 (1989), 1-13.  doi: 10.1016/0022-0396(89)90093-4.  Google Scholar

[19]

M. del Pino, R. Manásevich and A. E. Murúa, Existence and multiplicity of solutions with prescribed period for a second order quasilinear ODE, Nonlinear Anal., 18 (1992), 79-92.  doi: 10.1016/0362-546X(92)90048-J.  Google Scholar

[20]

M. del Pino and R. Manásevich, Infinitely many $2\pi$-periodic solutions for a problem arising in nonlinear elasticity, J. Differ. Equ., 103 (1993), 260-277.  doi: 10.1006/jdeq.1993.1050.  Google Scholar

[21]

D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach-Dancer condition, J. Differ. Equ., 171 (2001), 233-250.  doi: 10.1006/jdeq.2000.3847.  Google Scholar

[22]

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.  Google Scholar

[23]

D. Qian, L. Chen and X. Sun, Periodic solutions of superlinear impulsive differential equations: a geometric approach, J. Differ. Equ., 258 (2015), 3088-3106.  doi: 10.1016/j.jde.2015.01.003.  Google Scholar

[24]

D. Qian, P. J. Torres and P. Wang, Periodic solutions of second Order equations via rotation Numbers, J. Differ. Equ., 266 (2019), 4746–4768. doi: 10.1016/j.jde.2018.10.010.  Google Scholar

[25]

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.  Google Scholar

[26]

H. Royden, P. Fitzpatrick, Real Analysis, 4$^{th}$ edition, Printice-Hall Inc, Boston, 2010. Google Scholar

[27]

P. Yan and M. Zhang, Rotation number, periodic Fucik spectrum and multiple periodic solutions, Commun. Contemp. Math., 12 (2010), 437-455.  doi: 10.1142/S0219199710003877.  Google Scholar

[28]

M. Zhang, Nonuniform nonresonance at the first eigenvalue of the p-Laplacian, Nonlinear Anal., 29 (1997), 41-51.  doi: 10.1016/S0362-546X(96)00037-5.  Google Scholar

[29]

M. Zhang, Nonuniform nonresonance of semilinear differential equations, J. Differ. Equ., 166 (2000), 33-50.  doi: 10.1006/jdeq.2000.3798.  Google Scholar

Figure 1.  The trajectories in regions $ \mathcal{D}_{1} $ and $ \mathcal{D}_{2} $
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Figure 2.  Trajectory intersects $ y = 0 $ and $ y = -\delta $
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Figure 3.  Trajectory intersects $ y = 0 $ and $ y = \delta $
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