February  2020, 25(2): 555-567. doi: 10.3934/dcdsb.2019254

Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation

1. 

Department of Mathematics, University of Ruse, 7017 Ruse, Bulgaria

2. 

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

* Corresponding author: Nikolay Dimitrov

Received  September 2018 Revised  November 2018 Published  February 2020 Early access  November 2019

The aim of this paper is the study of existence of homoclinic solutions for a nonlinear difference equation involving $ p $-Laplacian. Under suitable growth conditions we prove that the considered problem has at least one homoclinic solution. The proof is based on the mountain-pass theorem with Cerami's condition, Brezis-Lieb lemma and variational method.

Citation: Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254
References:
[1]

C. J. Amick and J. F. Toland, Homoclinic orbits in the dynamic phase space analogy of an elastic strut, European J. Appl. Math., 3 (1992), 97-114.  doi: 10.1017/S0956792500000735.

[2]

P. Amster, P. De Nápoli and M. C. Mariani, Existence of solutions for elliptic systems with critical Sobolev exponent, ElectronicJournal of Differential Equations, 2002 (2002), 13 pp.

[3]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence functionals, Proc. Am. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.

[4]

B. Buffoni, Periodic and homoclinic orbits for Lorentz-Lagrangian systems via variational method, Nonlinear Anal., 26 (1996), 443-462.  doi: 10.1016/0362-546X(94)00290-X.

[5]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 19. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.

[6]

C.-Y. Li, Remarks on homoclinic solutions for semilinear fourth-order ordinary differential equations without periodicity, Appl. Math. J. Chinese Univ. Der. B, 24 (2009), 49-55.  doi: 10.1007/s11766-009-1948-z.

[7]

T. X. LiJ. T. Sun and T.-F. Wu, Existence of homoclinic solutions for a fourth order differential equation with a parameter, Appl. Math. and Comp., 251 (2015), 499-506.  doi: 10.1016/j.amc.2014.11.056.

[8]

L. A. Peletier and W. C. Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics, Progress in Nonlinear Differential Equations and their Applications, 45. Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0135-9.

[9]

L. Saavedra and S. Tersian, Existence of solutions for nonlinear $p$-Laplacian difference equations, Topological Methods in Nonlinear Analysis, 50 (2017), 151-167. 

[10]

L. Saavedra and S. Tersian, Existence of solutions for 2n-th order nonlinear $p$-Laplacian differential equations, Nonlinear Anal., 34 (2017), 507-519.  doi: 10.1016/j.nonrwa.2016.09.018.

[11]

J. T. Sun and T.-F. Wu, Two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation, J. Math. Anal. Appl., 413 (2014), 622-632.  doi: 10.1016/j.jmaa.2013.12.023.

[12]

S. Tersian and J. Chaparova, Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations, J. Math. Anal. Appl., 260 (2001), 490-506.  doi: 10.1006/jmaa.2001.7470.

show all references

References:
[1]

C. J. Amick and J. F. Toland, Homoclinic orbits in the dynamic phase space analogy of an elastic strut, European J. Appl. Math., 3 (1992), 97-114.  doi: 10.1017/S0956792500000735.

[2]

P. Amster, P. De Nápoli and M. C. Mariani, Existence of solutions for elliptic systems with critical Sobolev exponent, ElectronicJournal of Differential Equations, 2002 (2002), 13 pp.

[3]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence functionals, Proc. Am. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.

[4]

B. Buffoni, Periodic and homoclinic orbits for Lorentz-Lagrangian systems via variational method, Nonlinear Anal., 26 (1996), 443-462.  doi: 10.1016/0362-546X(94)00290-X.

[5]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 19. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.

[6]

C.-Y. Li, Remarks on homoclinic solutions for semilinear fourth-order ordinary differential equations without periodicity, Appl. Math. J. Chinese Univ. Der. B, 24 (2009), 49-55.  doi: 10.1007/s11766-009-1948-z.

[7]

T. X. LiJ. T. Sun and T.-F. Wu, Existence of homoclinic solutions for a fourth order differential equation with a parameter, Appl. Math. and Comp., 251 (2015), 499-506.  doi: 10.1016/j.amc.2014.11.056.

[8]

L. A. Peletier and W. C. Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics, Progress in Nonlinear Differential Equations and their Applications, 45. Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0135-9.

[9]

L. Saavedra and S. Tersian, Existence of solutions for nonlinear $p$-Laplacian difference equations, Topological Methods in Nonlinear Analysis, 50 (2017), 151-167. 

[10]

L. Saavedra and S. Tersian, Existence of solutions for 2n-th order nonlinear $p$-Laplacian differential equations, Nonlinear Anal., 34 (2017), 507-519.  doi: 10.1016/j.nonrwa.2016.09.018.

[11]

J. T. Sun and T.-F. Wu, Two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation, J. Math. Anal. Appl., 413 (2014), 622-632.  doi: 10.1016/j.jmaa.2013.12.023.

[12]

S. Tersian and J. Chaparova, Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations, J. Math. Anal. Appl., 260 (2001), 490-506.  doi: 10.1006/jmaa.2001.7470.

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