×

Multiple homoclinic solutions for second-order perturbed Hamiltonian systems. (English) Zbl 1294.37023

Here the authors consider the second-order perturbed Hamiltonian system \[ \ddot u(t)-\lambda L(t) u(t)+\nabla W(t,u(t))= f(t) \] for \(t\) real, \(u\in\mathbb{R}^n\), \(W\in C^1(\mathbb{R}\times \mathbb{R}^n,\mathbb{R})\) and where \(\nabla W(t,x)\) means the gradient with respect to \(x\). Further, \(L\in C(\mathbb{R},\mathbb{R}^{n^2})\) is a positive definite symmetric matrix for all real \(t\), and \(\lambda\geq 1\) is a parameter.
They prove that this equation has at least two nontrivial homoclinic solutions when \(f\in L^2(\mathbb{R},\mathbb{R}^n)\) is small enough and not identically zero, provided that \(W\) is either asymptotically quadratic or superquadratic in \(x\) as \(|x|\to\infty\).
The proofs of the main result use variational methods. The authors’ work differs from prior efforts that largely considered the cases where \(L\) and \(W\) are periodic in \(t\), or independent of \(t\).

MSC:

37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
70H09 Perturbation theories for problems in Hamiltonian and Lagrangian mechanics
70H30 Other variational principles in mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Carriao, Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems, J. Math. Anal. Appl. 230 pp 157– (1999) · Zbl 0919.34046 · doi:10.1006/jmaa.1998.6184
[2] Zelati, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 pp 693– (1991) · Zbl 0744.34045 · doi:10.2307/2939286
[3] Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal. 25 pp 1095– (1995) · Zbl 0840.34044 · doi:10.1016/0362-546X(94)00229-B
[4] Ding, Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal. 71 pp 1395– (2009) · Zbl 1168.58302 · doi:10.1016/j.na.2008.10.116
[5] Felmer, Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 pp 285– (1998) · Zbl 0919.58026
[6] Izydorek, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differ. Equ. 219 pp 375– (2005) · Zbl 1080.37067 · doi:10.1016/j.jde.2005.06.029
[7] Janczewska, Almost homoclinics for nonautonomous second order Hamiltonian systems by a variational approach, Bull. Belg. Math. Soc. Simon Stevin 17 pp 171– (2010) · Zbl 1230.34041
[8] Janczewska, Two almost homoclinic solutions for second-order perturbed Hamiltonian systems, Commun. Contemp. Math. 14 pp 1– (2012) · Zbl 1257.37041 · doi:10.1142/S0219199712500253
[9] Lv, Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials, Nonlinear Anal. Real World Appl. 13 pp 1152– (2012) · Zbl 1239.34046 · doi:10.1016/j.nonrwa.2011.09.008
[10] Lv, Existence of homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal. 72 pp 390– (2010) · Zbl 1186.34059 · doi:10.1016/j.na.2009.06.073
[11] Kryszewski, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ. 3 pp 441– (1998) · Zbl 0947.35061
[12] Omana, Homoclinic orbits for a class of Hamiltonian systems, Differ. Integral Equ. 5 pp 1115– (1992) · Zbl 0759.58018
[13] Ou, Existence of homoclinic solutions for the second order Hamiltonian systems, J. Math. Anal. Appl. 291 pp 203– (2004) · Zbl 1057.34038 · doi:10.1016/j.jmaa.2003.10.026
[14] Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 114 pp 33– (1990) · Zbl 0705.34054 · doi:10.1017/S0308210500024240
[15] Rabinowitz, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z. 206 pp 473– (1991) · Zbl 0707.58022 · doi:10.1007/BF02571356
[16] Tang, Homoclinic solutions for a class of second-order Hamiltonian systems, J. Math. Anal. Appl. 354 pp 539– (2009) · Zbl 1179.37082 · doi:10.1016/j.jmaa.2008.12.052
[17] Tang, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. R. Soc. Edinburgh Sec. A 141 pp 1103– (2011) · Zbl 1237.37044 · doi:10.1017/S0308210509001346
[18] Wang, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl. 366 pp 569– (2010) · Zbl 1204.37067 · doi:10.1016/j.jmaa.2010.01.060
[19] Wan, Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Discrete Contin. Dyn. Syst. Ser. B 15 pp 255– (2011)
[20] Yang, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials, Nonlinear Anal. Real World Appl. 12 pp 2742– (2011) · Zbl 1343.37052 · doi:10.1016/j.nonrwa.2011.03.019
[21] Yang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic potentials, Nonlinear Anal. RWA 10 pp 1417– (2009) · Zbl 1162.34328 · doi:10.1016/j.nonrwa.2008.01.013
[22] Zhang, Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal. 72 pp 894– (2010) · Zbl 1178.37063 · doi:10.1016/j.na.2009.07.021
[23] Zhang, Homoclinic solutions for a class of second order Hamiltionian systems with locally defined potentials, Nonlinear Anal. 75 pp 3188– (2012) · Zbl 1243.37054 · doi:10.1016/j.na.2011.12.018
[24] Y. Ye C.-L. Tang Existence and multiplicity results of homoclinic orbits for second order Hamiltonian systems
[25] Zhu, Existence of homoclinic solutions for a class of second order systems, Nonlinear Anal. 75 pp 2455– (2012) · Zbl 1285.34042 · doi:10.1016/j.na.2011.10.043
[26] Zou, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett. 16 pp 1283– (2003) · Zbl 1039.37044 · doi:10.1016/S0893-9659(03)90130-3
[27] Wang, Existence of infinitely many homoclinic orbits for nonperiodic superquadratic Hamiltonian systems, Nonlinear Anal. 75 pp 4873– (2012) · Zbl 1248.34055 · doi:10.1016/j.na.2012.04.002
[28] Ding, Homoclinic orbits for a nonperiodic Hamiltonian system, J. Differ. Equ. 237 pp 473– (2007) · Zbl 1117.37032 · doi:10.1016/j.jde.2007.03.005
[29] Ambrosetti, Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova 89 pp 177– (1993) · Zbl 0806.58018
[30] Chen, Fast homoclinic solutions for a class of damped vibration problems, Appl. Math. Comput. 219 pp 6053– (2013) · Zbl 1305.34064 · doi:10.1016/j.amc.2012.10.103
[31] Lv, Existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with superquadratic potential, Abstr. Appl. Anal. 2013 pp 1– (2013) · Zbl 1267.34074 · doi:10.1155/2013/328630
[32] Korman, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differ. Equ. 1 pp 1– (1994) · Zbl 0788.34042
[33] Bartsch, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr. 279 pp 1267– (2006) · Zbl 1117.58007 · doi:10.1002/mana.200410420
[34] Ekeland, Nonconvex minimization problems, Bull. Am. Math. Soc. 1 pp 443– (1979) · Zbl 0441.49011 · doi:10.1090/S0273-0979-1979-14595-6
[35] Ekeland, Convexity Methods in Hamiltonian Mechanics (1990) · Zbl 0707.70003 · doi:10.1007/978-3-642-74331-3
[36] Willem, Minimax Theorems (1996) · doi:10.1007/978-1-4612-4146-1
[37] Ding, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differ. Equ. 29 pp 397– (2007) · Zbl 1119.35082 · doi:10.1007/s00526-006-0071-8
[38] Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations (1986) · Zbl 0609.58002 · doi:10.1090/cbms/065
[39] Tang, Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl. 259 pp 386– (2001) · Zbl 0999.34039 · doi:10.1006/jmaa.2000.7401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.