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A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties
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 |
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 and Applied Analysis,
2021, 20
(5)
: 2117-2138.
doi: 10.3934/cpaa.2021060
![]()
References:
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Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear Anal., 10 (1986), 1083-1103.
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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.
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T. Ding and F. Zanolin,
Periodic solutions of Duffing's equations with superquadratic potential, J. Differ. Equ., 97 (1992), 328-378.
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Periodic solutions of weakly coupled superlinear systems, J. Differ. Equ., 260 (2016), 2150-2162.
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A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 679-698.
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J. Franks,
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M. García-Huidobro, R. Manásevich and F. Zanolin,
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On boundary value problems for superlinear second order differential equations, J. Differ. Equ., 26 (1977), 37-53.
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H. Jacobowitz,
Periodic solutions of $x''+f(t, x) = 0$ via the Poincaré-Birkhoff theorem, J. Differ. Equ., 20 (1976), 37-52.
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V. Lakshmikantham and S. Leela, Differential and integral inequalities; theory and applications, Academic Press, New York and London, 1969.
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R. Manásevich and J. Mawhin,
Periodic solutions for nonlinear systems with $p$-Laplacian like operators, J. Differ. Equ., 145 (1998), 367-393.
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| [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. 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. |
| [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 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. |
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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. |
| [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. 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. |
| [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. 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. |
| [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-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. |
| [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. 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. |
| [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 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. |
| [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. |
| [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. 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. |
| [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. |
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