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

Nikolay Dimitrov; Stepan Tersian

doi:10.3934/dcdsb.2019254

Article Contents

2020, Volume 25, Issue 2: 555-567. Doi: 10.3934/dcdsb.2019254

This issue Previous Article Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach Next Article Well-posedness results for fractional semi-linear wave equations

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
^* Corresponding author: Nikolay Dimitrov

Received: September 2018

Revised: November 2018

Early access: November 2019

Published: February 2020

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 34B15, 34C37; Secondary: 39A10.

Citation:

Full Text(HTML)

Related Papers

Cited by

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. 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.
[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.