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

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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

Abstract
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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 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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