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

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

Nikolay Dimitrov 1,, and  Stepan Tersian 2,

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.

Keywords: Fourth order difference equation, homoclinic solutions, mountain pass theorem, variational method.
Mathematics Subject Classification: Primary: 34B15, 34C37; Secondary: 39A10.
Citation: Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[7]

T. X. Li, J. 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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

T. X. Li, J. 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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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