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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

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}$.
Citation: Li-Li Wan, Chun-Lei Tang. Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[17]

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

[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.

[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.

[20]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[17]

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

[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.

[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.

[20]

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

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