Homoclinic Solutions for Some Nonperiodic Fourth Order Differential Equations with Sublinear Nonlinearities
DOI:
https://doi.org/10.15377/2409-5761.2017.04.3Keywords:
Homoclinic solutions, Critical point, Variational methods, GenusAbstract
In this paper we investigate the existence of homoclinic solutions for the following fourth order nonautonomous differential equations; u(4) + wu’’ + a(x)u = f (x,u), (FDE) where w is a constant, a ɛ C(R, R) and f ɛ C(R x R, R) . The novelty of this paper is that, when (FDE) is nonperiodic, i.e., a and f are nonperiodic in x, assuming that a is bounded from below and f is sublinear as | u |→ +ꚙ , we establish one new criterion to guarantee the existence and multiplicity of homoclinic solutions of (FDE). Recent results in the literature are generalized and improved.
References
Amick CJ and Toland JF. Homoclinic orbits in the dynamic phase space analogy of an elastic strut, European. J Appl Math 1992; 3: 97-114. https://doi.org/10.1017/S0956792500000735 DOI: https://doi.org/10.1017/S0956792500000735
Bretherton FP. Resonant interaction between waves: the case of discrete oscillations. J Fluid Mech 1964; 20: 457-479. https://doi.org/10.1017/S0022112064001355 DOI: https://doi.org/10.1017/S0022112064001355
Buffoni B. Periodic and homoclinic orbits for Lorentz- Lagrangian systems via variational method. Nonlinear Anal 1996; 26: 443-462. https://doi.org/10.1016/0362-546X(94)00290-X DOI: https://doi.org/10.1016/0362-546X(94)00290-X
Buffoni B, Groves M and Toland JF. A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers. Philos Trans Roy Soc London Ser A 1996; 354(1707): 575-607. https://doi.org/10.1098/rsta.1996.0020 DOI: https://doi.org/10.1098/rsta.1996.0020
Buryak AV and Kivshar Y. Solitons due to second harmonic generation. Phys Lett A 1995; 197: 407-412. https://doi.org/10.1016/0375-9601(94)00989-3 DOI: https://doi.org/10.1016/0375-9601(94)00989-3
Chapiro G, Faria LFO and Maldonado AD. On the existence of solutions for a class of fourth order differential equations. J Math Anal Appl 2015; 427: 126-139. https://doi.org/10.1016/j.jmaa.2015.01.012 DOI: https://doi.org/10.1016/j.jmaa.2015.01.012
Coullet P, Elphick C and Repaux D. Nature of spatial chaos. Phys Rev Lett 1987; 58: 431-434. https://doi.org/10.1103/PhysRevLett.58.431 DOI: https://doi.org/10.1103/PhysRevLett.58.431
Dee GT and van Saarloos W. Bistable systems with propagating fronts leading to pattern formation. Phys Rev Lett 1988; 60: 2641-2644. https://doi.org/10.1103/PhysRevLett.60.2641 DOI: https://doi.org/10.1103/PhysRevLett.60.2641
Lega J, Molonev J and Newell A. Swift-Hohenberg for lasers. Phys Rev Lett 1994; 73: 2978-2981. https://doi.org/10.1103/PhysRevLett.73.2978 DOI: https://doi.org/10.1103/PhysRevLett.73.2978
Li F, Sun JT, Lu GF and Lv CJ. Infinitely many homoclinic solutions for a nonperiodic fourth-order differential equation without (AR)-condition. Appl Math Comput 2014; 241: 36-41. https://doi.org/10.1016/j.amc.2014.04.067 DOI: https://doi.org/10.1016/j.amc.2014.04.067
Li TX, Sun JT and Wu TF. Existence of homoclinic solutions for a fourth order differential equation with a parameter. Appl Math Comput 2015; 251: 499-506. https://doi.org/10.1016/j.amc.2014.11.056 DOI: https://doi.org/10.1016/j.amc.2014.11.056
Li CY. Homoclinic orbits of two classes of fourth order semilinear differential equations with periodic nonlinearity. J Appl Math Comput 2008; 27: 107-116. https://doi.org/10.1007/s12190-008-0045-4 DOI: https://doi.org/10.1007/s12190-008-0045-4
Li CY. Remarks on homoclinic solutions for semilinear fourthorder differential equations without periodicity. Appl Math J Chinese Univ 2009; 24: 49-55. https://doi.org/10.1007/s11766-009-1948-z DOI: https://doi.org/10.1007/s11766-009-1948-z
Peletier LA and Troy WC. Spatial Patterns: Higher Order Models in Physics and Mechnics, Birkhäuser, Boston, 2001. https://doi.org/10.1007/978-1-4612-0135-9 DOI: https://doi.org/10.1007/978-1-4612-0135-9
Rabinowitz PH. Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics. The American Mathematical Society, Providence, RI, 1986; 65. https://doi.org/10.1090/cbms/065 DOI: https://doi.org/10.1090/cbms/065
Salvatore A. Homoclinic orbits for a special class of nonautonomous Hamiltonian systems. Nonlinear Anal 1997; 30: 4849-4857. https://doi.org/10.1016/S0362-546X(97)00142-9 DOI: https://doi.org/10.1016/S0362-546X(97)00142-9
Sun JT and Wu TF. Two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation. J Math Anal Appl 2014; 413: 622-632. https://doi.org/10.1016/j.jmaa.2013.12.023 DOI: https://doi.org/10.1016/j.jmaa.2013.12.023
Sun JT, Wu TF and Li F. Concentration of homoclinic solutions for some fourth-order equations with sublinear indefinite nonlinearities. Appl Math Lett 2014; 38: 1-6. https://doi.org/10.1016/j.aml.2014.06.009 DOI: https://doi.org/10.1016/j.aml.2014.06.009
Tersian S and Chaparova J. Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations. J Math Anal Appl 2001; 260: 490-506. https://doi.org/10.1006/jmaa.2001.7470 DOI: https://doi.org/10.1006/jmaa.2001.7470
Willem M. Minimax Theorems, Progr. Nonlinear Differential Equations Appl, Birkhäuser, Boston, 1996; 24. DOI: https://doi.org/10.1007/978-1-4612-4146-1
Yang L. Infinitely many homoclinic solutions for nonperiodic fourth order differential equations with general potentials. Abst Appl Anal 2014, Art. ID 435125, 7 pp. DOI: https://doi.org/10.1155/2014/435125
Yang L. Multiplicity of homoclinic solutions for a class of nonperiodic fourth order differential equations with general perturbation. Abstr Appl Anal 2014; Art. ID 126435, 5 pp. DOI: https://doi.org/10.1155/2014/126435
Zhang ZH and Yuan R. Homoclinic solutions for a nonperiodic fourth order differential equations without coercive conditions. Appl Math Comput 2015; 250: 280-286. https://doi.org/10.1016/j.amc.2014.10.114 DOI: https://doi.org/10.1016/j.amc.2014.10.114
Downloads
Published
Issue
Section
License
Copyright (c) 2017 Journal of Advances in Applied & Computational Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
All the published articles are licensed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC 4.0) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.
