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  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 & 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

[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. 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.  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. 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.  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

[1]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[2]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[3]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[4]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[5]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[6]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[7]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1

[8]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

[9]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065

[10]

Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511

[11]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133

[12]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[13]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[14]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[15]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[16]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[17]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[18]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[19]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[20]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (192)
  • HTML views (121)
  • Cited by (1)

Other articles
by authors

[Back to Top]