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

Periodic solutions of p-Laplacian equations via rotation numbers

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  May 2021 Early access  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.

Citation: Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060
References:
[1]

L. BoccardoP. DràbekD. 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[10]

M. García-HuidobroR. 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.

[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.

[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.

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

[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.

[16]

A. MargheriC. 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.

[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.

[18]

M. del PinoM. 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.

[19]

M. del PinoR. 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.

[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.

[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.

[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.

[23]

D. QianL. 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.

[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.

[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.

[26]

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

[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.

[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.

[29]

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

show all references

References:
[1]

L. BoccardoP. DràbekD. 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[10]

M. García-HuidobroR. 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.

[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.

[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.

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

[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.

[16]

A. MargheriC. 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.

[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.

[18]

M. del PinoM. 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.

[19]

M. del PinoR. 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.

[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.

[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.

[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.

[23]

D. QianL. 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.

[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.

[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.

[26]

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

[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.

[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.

[29]

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

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