Solitary wave and periodic wave solutions for a class of singular p-laplacian systems with impulsive effects
Abstract
This work deals with the existence of periodic wave solutions and nonexistence of solitary wave solutions for a class of second-order singular p-Laplacian systems with impulsive effects. A su_cient criterion for the solutions of the considered system is provided via an innovative method of the mountain pass theorem and an approximation technique. Some corresponding results in the literature can be enriched and extended.
Keyword : periodic wave solution, solitary wave solution, singular p-Laplacian systems, impulsive effects, mountain pass theorem
How to Cite
Kong, F., Luo, Z., & Qiu, H. (2018). Solitary wave and periodic wave solutions for a class of singular p-laplacian systems with impulsive effects. Mathematical Modelling and Analysis, 23(1), 17-32. https://doi.org/10.3846/mma.2018.002
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Sun and D. O’Regan. Impulsive periodic solution for singular problems via variational methods. Bull. Aust. Math. Soc., 86(2):193–204, 2012. https://doi.org/10.1017/S0004972711003509
Y. Tian and W. Ge. Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc., 51(2):509–527, 2008. https://doi.org/10.1017/S0013091506001532
Y. Tian and W. Ge. Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations. Nonlinear Anal., 72(1):277–287, 2010. https://doi.org/10.1016/j.na.2009.06.051
Z. Wang. Periodic solutions of Li´enard equation with a singularity and a deviating argument. Nonlinear Anal. Real World Appl., 16:227–234, 2014. https://doi.org/10.1016/j.nonrwa.2013.09.021
Z. Wang and T. Ma. Existence and multiplicity of periodic solutions of semilinear resonant Duffing equations with singularities. Nonlinearity, 25(2):279–307, 2012. https://doi.org/10.1088/0951-7715/25/2/279
J. Xie and Z. Luo. Homoclinic orbits for Hamiltonian systems induced by impulses. Math. Methods Appl. Sci., 39(9):2239–2250, 2016. https://doi.org/10.1002/mma.3636
G. Yang, J. Lu and A. Luo. On the computation of Lyapunov exponents for forced vibration of a Lennard-Jones oscillator. Chaos Solitons Fractals, 23(3):833–841, 2005. https://doi.org/10.1016/j.chaos.2004.05.034
S. Zavalishchin and A. Sesekin. Dynamic Impulse Systems: Theory and Applications. Kluwer Academic Publishers, United States, 1997. https://doi.org/10.1007/978-94-015-8893-5
H. Zhang and Z. Li. Periodic and homoclinic solutions generated by impulses. Nonlinear Anal. Real World Appl., 12(1):39–51, 2011. https://doi.org/10.1016/j.nonrwa.2010.05.034
M. Zhang. Periodic solutions of damped differential systems with repulsive singular forces. Proc. Amer. Math. Soc., 127:401–407, 1999. https://doi.org/10.1090/S0002-9939-99-05120-5
M. Benchohra, J. Henderson and S. Ntouyas. Impulsive Differential Equations and Inclusions, volume 2. Hindawi Publishing Corporation, New York, 2006. https://doi.org/10.1155/9789775945501
D. Bonheure and P. Torres. Bounded and homoclinic-like solutions of a second-order singular differential equation. Bull. Lond. Math. Soc., 44(1):47–54, 2012. https://doi.org/10.1112/blms/bdr060
L. Chen, C. Tisdell and R. Yuan. On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl., 331(2):233–244, 2007. https://doi.org/10.1016/j.jmaa.2006.09.021
A. Fonda and R. Toader. Periodic orbits of radially symmetric Keplerianlike systems: A topological degree approach. J. Differential Equations., 244(12):3235–3264, 2008. https://doi.org/10.1016/j.jde.2007.11.005
D. Franco and P. Torres. Periodic solutions of singular systems without the strong force condition. Proc. Amer. Math. Soc., 136:1229–1236, 2008. https://doi.org/10.1090/S0002-9939-07-09226-X
R. Hakl and P. Torres. On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Differential Equations., 248(1):111–126, 2010. https://doi.org/10.1016/j.jde.2009.07.008
X. Han and H. Zhang. Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system. J. Comput. Appl.Math., 235(5):1531–1541, 2011. https://doi.org/10.1016/j.cam.2010.08.040
A. Huaux. Sur l’existence d’une solution p´eriodique de l’e quation diff´erentielle non lin´eaire x¨ + (0, 2)x˙ + 1−xx = (0, 5) cos ωt. Bull. Acad. r. Belg., 48:494–504, 1962.
M. Izydorek and J. Janczewska. Homoclinic solutions for a class of the second order Hamiltonian systems. J. Differential Equations., 219(2):375–389, 2005. https://doi.org/10.1016/j.jde.2005.06.029
F. Kong, S. Lu and Z. Luo. Positive periodic solutions for singular higher order delay differential equations. Results Math., 72(1–2):71–86, 2017. https://doi.org/10.1007/s00025-016-0647-3
F. Kong and Z. Luo. Positive periodic solutions for a kind of first-order singular differential equation induced by impulses. Qual. Theory Dyn. Syst., 2017. https://doi.org/10.1007/s12346-017-0239-y
F. Kong, Z. Luo and F. Chen. Solitary wave solutions for singular non-Newtonian filtration equations. J. Math. Phys., 58(9):093506, 2017. https://doi.org/10.1063/1.5005100
P. Lindqvist. On the equation ÷(|∇u|p−2∇u) + λ|u|p−2u = 0. Proc. Edinb.Math. Soc., 109:157–164, 1990.
J. Mawhin. Topological degree and boundary value problems for nonlinear differential equations. In In: Furi, M., Zecca, P. (eds.) Topologic Methods for Ordinary Differential Equations, volume 1537 of Lecture Notes in Mathematics, pp. 74–142, New York, 1993. Springer. https://doi.org/10.1007/BFb0085076
J. Mawhin and M. Willem. Critical Point Theory and Hamiltonian Systems. Springer, 1989. https://doi.org/10.1007/978-1-4757-2061-7
J. Nieto and D. O’Regan. Variational approach to impulsive differential equations. Nonlinear Anal. Real World Appl., 10(2):680–690, 2009. https://doi.org/10.1016/j.nonrwa.2007.10.022
H. Pishkenari, M. Behzad and A. Meghdari. Nonlinear dynamic analysis of atomic force microscopy under deterministic and random excitation. Chaos Solitons Fractals, 37(3):748–762, 2008. https://doi.org/10.1016/j.chaos.2006.09.079
D. Qian and X. Li. Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl., 303(1):288–303, 2005. https://doi.org/10.1016/j.jmaa.2004.08.034
I. Rachůnková and J. Tomeček. Homoclinic solutions of singular nonautonomous second-order differential equations. Bound. Value Probl., 959636:1–21, 2009. https://doi.org/10.1155/2009/959636
S. Rutzel, S. Lee and A. Raman. Nonlinear dynamics of atomic-force-microscope probes driven in Lennard-Jones potentials. Proc. Roy. Soc. Edinburgh Sect. A., 459(2036):1925–1948, 2003. https://doi.org/10.1098/rspa.2002.1115
A. Samoilenko and N. Perestyuk. Impulsive Differential Equations. World Scientific, Singapore, 1989.
J. Sun, J. Chu and H. Chen. Periodic solution generated by impulses for singular differential equations. J. Math. Anal. Appl., 404(2):562–569, 2013. https://doi.org/10.1016/j.jmaa.2013.03.036
J. Sun and D. O’Regan. Impulsive periodic solution for singular problems via variational methods. Bull. Aust. Math. Soc., 86(2):193–204, 2012. https://doi.org/10.1017/S0004972711003509
Y. Tian and W. Ge. Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc., 51(2):509–527, 2008. https://doi.org/10.1017/S0013091506001532
Y. Tian and W. Ge. Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations. Nonlinear Anal., 72(1):277–287, 2010. https://doi.org/10.1016/j.na.2009.06.051
Z. Wang. Periodic solutions of Li´enard equation with a singularity and a deviating argument. Nonlinear Anal. Real World Appl., 16:227–234, 2014. https://doi.org/10.1016/j.nonrwa.2013.09.021
Z. Wang and T. Ma. Existence and multiplicity of periodic solutions of semilinear resonant Duffing equations with singularities. Nonlinearity, 25(2):279–307, 2012. https://doi.org/10.1088/0951-7715/25/2/279
J. Xie and Z. Luo. Homoclinic orbits for Hamiltonian systems induced by impulses. Math. Methods Appl. Sci., 39(9):2239–2250, 2016. https://doi.org/10.1002/mma.3636
G. Yang, J. Lu and A. Luo. On the computation of Lyapunov exponents for forced vibration of a Lennard-Jones oscillator. Chaos Solitons Fractals, 23(3):833–841, 2005. https://doi.org/10.1016/j.chaos.2004.05.034
S. Zavalishchin and A. Sesekin. Dynamic Impulse Systems: Theory and Applications. Kluwer Academic Publishers, United States, 1997. https://doi.org/10.1007/978-94-015-8893-5
H. Zhang and Z. Li. Periodic and homoclinic solutions generated by impulses. Nonlinear Anal. Real World Appl., 12(1):39–51, 2011. https://doi.org/10.1016/j.nonrwa.2010.05.034
M. Zhang. Periodic solutions of damped differential systems with repulsive singular forces. Proc. Amer. Math. Soc., 127:401–407, 1999. https://doi.org/10.1090/S0002-9939-99-05120-5