Topological Methods in Nonlinear Analysis

Existence of solutions for nonlinear $p$-Laplacian difference equations

Lorena Saavedra and Stepan Tersian

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The aim of this paper is the study of existence of solutions for nonlinear $2n^{\rm th}$-order difference equations involving $p$-Laplacian. In the first part, the existence of a nontrivial homoclinic solution for a discrete $p$-Laplacian problem is proved. The proof is based on the mountain-pass theorem of Brezis and Nirenberg. Then, we study the existence of multiple solutions for a discrete $p$-Laplacian boundary value problem. In this case the proof is based on the three critical points theorem of Averna and Bonanno.

Article information

Topol. Methods Nonlinear Anal., Volume 50, Number 1 (2017), 151-167.

First available in Project Euclid: 14 October 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Saavedra, Lorena; Tersian, Stepan. Existence of solutions for nonlinear $p$-Laplacian difference equations. Topol. Methods Nonlinear Anal. 50 (2017), no. 1, 151--167. doi:10.12775/TMNA.2017.022.

Export citation


  • R.P. Agarwal, K. Perera and D. O' Regan, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (2004), 69–73.
  • R.P. Agarwal and F. Wong, Existence of positive solutions for higher order difference equations, Appl. Math. Lett. 10 (1997), no. 5, 67–74.
  • F.M. Atici and A. Cabada, Existence and uniqueness results for discrete second-order periodic boundary value problems, Advances in Difference Equations IV, Comput. Math. Appl. 45 (2003), no. 6–9, 1417–1427.
  • D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. Methods Nonlinear Anal. 22 (2003), 93–103.
  • H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8–9, 939–963.
  • A. Cabada, C. Li and S. Tersian, On homoclinic solutions of a semilinear $p$-Laplacian difference equation with periodic coefficients, Adv. Difference Equations (2010), 17 pp.
  • A. Cabada, A. Iannizzott and S. Tersian, Existence of solutions of discrete equations via critical point theory, Proceedings of the International Workshop Future Directions in Difference Equations, Vigo, Spain, 2011, pp. 61–75.
  • X. Cai and J. Yu, Existence of periodic solutions for a $2n^{\mathrm{th}}$-order nonlinear difference equation, J. Math. Anal. Appl. 329 (2007), 870–878.
  • P. Chen and X. Tang, Existence of homoclinic orbits for $2n^{\mathrm{th}}$-order nonlinear difference equations containing both many advances and retardations, J. Math. Anal. Appl. 381 (2011), 485–505.
  • P. Candito and N. Giovannelli, Multiple solutions for a discrete boundary value problem involving the $p$-Laplacian, Comput. Math. Appl. 56 (2008), 959–964.
  • N. Dimitrov, Multiple solutions for a nonlinear discrete fourth order $p$- Laplacian equation, Proceedings of Union of Scientists, Ruse, 13 (2016), 16–25.
  • Z.M. Guo and J.S. Yu, The existence of periodic and subharmonic solutions of sub-quadratic second order difference equations, J. London Math. Soc. (2) 68 (2003), 419–430.
  • Z. He, On the existence of positive solutions of $p$-Laplacian difference equations, J. Comput. Appl. Math. 161 (2003), 193–201.
  • A. Ianizzotto and S. Tersian, Multiple homoclinic solutions for the discrete $p$-Laplacian via critical point theory, J. Math. Anal. Appl. 403 (2013), 173–182.
  • X. Liu, Y. Zhang and H. Shi, Periodic and subharmonic solutions for $2n$ th-order $p$-Laplacian difference equations, J. Contemp. Math. Anal. 49 (2014), 223–231.
  • M. Mihailescu, V.D. Radulescu and S. Tersian, Homoclinic solutions of difference equations with variable exponents, Topol. Methods Nonlinear Anal. 38 (2011), 277–289.
  • L.A. Peletier and W.C. Troy, Spatial Patterns, Higher Order Models in Physics and Mechanics, Birkhäser, 2001.
  • L. Saavedra and S. Tersian, Existence of solutions for 2n th-order nonlinear p-Laplaciandifferential equations, Nonlinear Anal. 34 (2017), 507–519.
  • S. Tersian and J. Chaparova, Periodic and homoclinic solutions of extended Fisher–Kolmogorov equations, J. Math. Anal. Appl. 260 (2001), 490–506.
  • X.H. Tang, X. Lin and L. Xiao, Homoclinic solutions for a class of second order discrete Hamiltonian systems, J. Difference Equ. Appl. 16 (2010), 1257–1273.
  • D. Wang and W. Guan, Three positive solutions of boundary value problems for $p$-Laplacian difference equations, Comput. Math. Appl. 55 (2008), 1943–1949.