American Institute of Mathematical Sciences

Advanced
January  2018, 23(1): 487-492. doi: 10.3934/dcdsb.2018033

On homoclinic solutions for a second order difference equation with p-Laplacian

Robert Stegliński

Institute of Mathematics, Lodz University of Technology, Wolczanska 215, 90-924 Lodz, Poland

Received  July 2016 Revised  September 2016 Published  January 2018

In this paper, we obtain conditions under which the difference equation
$-Δ ≤ft( a(k)φ _{p}(Δ u(k-1))) +b(k)φ_{p}(u(k))=λ f(k, u(k)), \;\;k∈\mathbb{Z}, $
has infinitely many homoclinic solutions. A variant of the fountain theorem is utilized in the proof of our theorem. Some known results in the literature are extended and complemented.
Keywords: Difference equations, discrete p-Laplacian, variational methods, infinitely many solutions, fountain theorem.
Mathematics Subject Classification: Primary: 39A10, 34C37; Secondary: 35B38.
Citation: Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033
References:
[1]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009 (2009), Art. ID 670675, 20 pp.  Google Scholar

[2]

A. Iannizzotto and S. Tersian, Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory, J. Math. Anal. Appl., 403 (2013), 173-182.  doi: 10.1016/j.jmaa.2013.02.011.  Google Scholar

[3]

L. Kong, Homoclinic solutions for a second order difference equation with p-Laplacian, Appl. Math. Comput., 247 (2014), 1103-1121.  doi: 10.1016/j.amc.2014.09.069.  Google Scholar

[4]

L. Kong, Homoclinic solutions for a higher order difference equation with p-Laplacian, Indag. Math., 27 (2016), 124-146.  doi: 10.1016/j.indag.2015.08.007.  Google Scholar

[5]

S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016.  Google Scholar

[6]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410.  doi: 10.1016/S0377-0427(99)00269-1.  Google Scholar

[7]

R. Stegliński,, On sequences of large homoclinic solutions for a difference equations on the integers, Adv. Difference Equ., 2016 (2016), 11pp.  doi: 10.1186/s13662-017-1344-6.  Google Scholar

[8]

R. Stegliński, On sequences of large homoclinic solutions for a difference equations on the integers involving oscillatory nonlinearities, Electron. J. Qual. Theory Differ. Equ., 35 (2016), 1-11.   Google Scholar

[9]

G. Sun and A. Mai, Infinitely many homoclinic solutions for second order nonlinear difference equations with p-Laplacian The Scientific World Journal 2014 (2014), Article ID 276372, 6 pages. doi: 10.1155/2014/276372.  Google Scholar

show all references

References:
[1]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009 (2009), Art. ID 670675, 20 pp.  Google Scholar

[2]

A. Iannizzotto and S. Tersian, Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory, J. Math. Anal. Appl., 403 (2013), 173-182.  doi: 10.1016/j.jmaa.2013.02.011.  Google Scholar

[3]

L. Kong, Homoclinic solutions for a second order difference equation with p-Laplacian, Appl. Math. Comput., 247 (2014), 1103-1121.  doi: 10.1016/j.amc.2014.09.069.  Google Scholar

[4]

L. Kong, Homoclinic solutions for a higher order difference equation with p-Laplacian, Indag. Math., 27 (2016), 124-146.  doi: 10.1016/j.indag.2015.08.007.  Google Scholar

[5]

S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016.  Google Scholar

[6]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410.  doi: 10.1016/S0377-0427(99)00269-1.  Google Scholar

[7]

R. Stegliński,, On sequences of large homoclinic solutions for a difference equations on the integers, Adv. Difference Equ., 2016 (2016), 11pp.  doi: 10.1186/s13662-017-1344-6.  Google Scholar

[8]

R. Stegliński, On sequences of large homoclinic solutions for a difference equations on the integers involving oscillatory nonlinearities, Electron. J. Qual. Theory Differ. Equ., 35 (2016), 1-11.   Google Scholar

[9]

G. Sun and A. Mai, Infinitely many homoclinic solutions for second order nonlinear difference equations with p-Laplacian The Scientific World Journal 2014 (2014), Article ID 276372, 6 pages. doi: 10.1155/2014/276372.  Google Scholar

[1]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[2]

Petru Jebelean. Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267
[3]

Xiying Sun, Qihuai Liu, Dingbian Qian, Na Zhao. Infinitely many subharmonic solutions for nonlinear equations with singular $ \phi $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 279-292. doi: 10.3934/cpaa.20200015
[4]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4805-4821. doi: 10.3934/dcds.2021058
[5]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922
[6]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[7]

Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593
[8]

Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060
[9]

John R. Graef, Lingju Kong, Min Wang. Existence of homoclinic solutions for second order difference equations with $p$-laplacian. Conference Publications, 2015, 2015 (special) : 533-539. doi: 10.3934/proc.2015.0533
[10]

Magdalena Nockowska-Rosiak, Piotr Hachuła, Ewa Schmeidel. Existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 369-375. doi: 10.3934/dcdsb.2018025
[11]

Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371
[12]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[13]

Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure & Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019
[14]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083
[15]

Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063
[16]

Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595
[17]

Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191
[18]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
[19]

Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071
[20]

Liang Zhang, X. H. Tang, Yi Chen. Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 823-842. doi: 10.3934/cpaa.2017039

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (75)
  • HTML views (120)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences