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January  2011, 15(1): 255-271. doi: 10.3934/dcdsb.2011.15.255

Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition

Li-Li Wan 1, and  Chun-Lei Tang 1,

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China

Received  June 2009 Revised  September 2010 Published  October 2010

The existence and multiplicity of homoclinic orbits for a class of the second order Hamiltonian systems $\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \forall t \in \mathbb{R}$, are obtained via the concentration-compactness principle and the fountain theorem respectively, where $W(t, x)$ is superquadratic and need not satisfy the (AR) condition with respect to the second variable $ x\in\mathbb{R}^{N}$.
Keywords: concentration-compactness principle, fountain theorem., Homoclinic orbits, second order Hamiltonian systems.
Mathematics Subject Classification: Primary: 34C37, 37J45; Secondary: 47J3.
Citation: Li-Li Wan, Chun-Lei Tang. Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 255-271. doi: 10.3934/dcdsb.2011.15.255
References:
[1]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16 (2003), 639-642. doi: doi:10.1016/S0893-9659(03)00059-4.  Google Scholar

[2]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: doi:10.1016/0362-546X(83)90115-3.  Google Scholar

[3]

V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: doi:10.1090/S0894-0347-1991-1119200-3.  Google Scholar

[4]

Y. H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113. doi: doi:10.1016/0362-546X(94)00229-B.  Google Scholar

[5]

Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math., 8 (2006), 453-480. doi: doi:10.1142/S0219199706002192.  Google Scholar

[6]

M. F. Furtado, L. A. Maia and E. A. B. Silva, Systems with coupling in $R^N$ for a class of noncoercive potentials, Discrete Contin. Dyn. Syst., 2003, suppl., 295-304.  Google Scholar

[7]

P. L. Felmer and E. A. B. Silva, Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285-301.  Google Scholar

[8]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, (1994), approx. 10 pp.  Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin, 1995.  Google Scholar

[10]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  Google Scholar

[11]

Y. Lv and C. L. Tang, Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198. doi: doi:10.1016/j.na.2006.08.043.  Google Scholar

[12]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.  Google Scholar

[13]

Z. Q. Ou and C. L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213. doi: doi:10.1016/j.jmaa.2003.10.026.  Google Scholar

[14]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  Google Scholar

[15]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: doi:10.1007/BF02571356.  Google Scholar

[16]

P. H. Rabinowitz, Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. Variations, 1 (1993), 1-36. doi: doi:10.1007/BF02163262.  Google Scholar

[17]

M. Willem, "Minimax Theorems," 24, Birkhäuser Boston, Inc., Boston, MA, 1996.  Google Scholar

[18]

J. Yang and F. B. Zhao, Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials, Nonlinear Analysis: Real World Applications, 10 (2009), 1417-1423. doi: doi:10.1016/j.nonrwa.2008.01.013.  Google Scholar

[19]

W. M. Zou and S. J. Li, Infinitely many homoclinic orbits for the second-order hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287. doi: doi:10.1016/S0893-9659(03)90130-3.  Google Scholar

[20]

W. M. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358. doi: doi:10.1007/s002290170032.  Google Scholar

show all references

References:
[1]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16 (2003), 639-642. doi: doi:10.1016/S0893-9659(03)00059-4.  Google Scholar

[2]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: doi:10.1016/0362-546X(83)90115-3.  Google Scholar

[3]

V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: doi:10.1090/S0894-0347-1991-1119200-3.  Google Scholar

[4]

Y. H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113. doi: doi:10.1016/0362-546X(94)00229-B.  Google Scholar

[5]

Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms, Commun. Contemp. Math., 8 (2006), 453-480. doi: doi:10.1142/S0219199706002192.  Google Scholar

[6]

M. F. Furtado, L. A. Maia and E. A. B. Silva, Systems with coupling in $R^N$ for a class of noncoercive potentials, Discrete Contin. Dyn. Syst., 2003, suppl., 295-304.  Google Scholar

[7]

P. L. Felmer and E. A. B. Silva, Homoclinic and periodic orbits for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285-301.  Google Scholar

[8]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, (1994), approx. 10 pp.  Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin, 1995.  Google Scholar

[10]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  Google Scholar

[11]

Y. Lv and C. L. Tang, Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198. doi: doi:10.1016/j.na.2006.08.043.  Google Scholar

[12]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.  Google Scholar

[13]

Z. Q. Ou and C. L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213. doi: doi:10.1016/j.jmaa.2003.10.026.  Google Scholar

[14]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  Google Scholar

[15]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: doi:10.1007/BF02571356.  Google Scholar

[16]

P. H. Rabinowitz, Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. Variations, 1 (1993), 1-36. doi: doi:10.1007/BF02163262.  Google Scholar

[17]

M. Willem, "Minimax Theorems," 24, Birkhäuser Boston, Inc., Boston, MA, 1996.  Google Scholar

[18]

J. Yang and F. B. Zhao, Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials, Nonlinear Analysis: Real World Applications, 10 (2009), 1417-1423. doi: doi:10.1016/j.nonrwa.2008.01.013.  Google Scholar

[19]

W. M. Zou and S. J. Li, Infinitely many homoclinic orbits for the second-order hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287. doi: doi:10.1016/S0893-9659(03)90130-3.  Google Scholar

[20]

W. M. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358. doi: doi:10.1007/s002290170032.  Google Scholar

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