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Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach
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 |
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.
References:
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Homoclinic orbits in the dynamic phase space analogy of an elastic strut, European J. Appl. Math., 3 (1992), 97-114.
doi: 10.1017/S0956792500000735. |
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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. |
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Periodic and homoclinic orbits for Lorentz-Lagrangian systems via variational method, Nonlinear Anal., 26 (1996), 443-462.
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I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 19. Springer-Verlag, Berlin, 1990.
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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. |
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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. |
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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. |
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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. |
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. |
| [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. |
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