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September  2014, 34(9): 3353-3369. doi: 10.3934/dcds.2014.34.3353

Homoclinic orbits of first-order superquadratic Hamiltonian systems

Cyril Joel Batkam 1,

1. 

Département de Mathématiques, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1, Canada

Received  May 2013 Revised  November 2013 Published  March 2014

In this article, we study the existence of homoclinic orbits for the first-order Hamiltonian system \begin{equation*} J\dot{u}(t)+\nabla H(t,u(t))=0,\quad t\in\mathbb{R}. \end{equation*} Under the Ambrosetti-Rabinowitz's superquadraticy condition, or no Ambrosetti-Rabinowitz's superquadracity condition, we present two results on the existence of infinitely many large energy homoclinic orbits when $H$ is even in $u$. We apply the generalized (variant) fountain theorems established recently by the author and Colin. Under no Ambrosetti-Rabinowitz's superquadracity condition, we also obtain the existence of a ground state homoclinic orbit by using the method of the generalized Nehari manifold for strongly indefinite functionals developed by Szulkin and Weth.
Keywords: homoclinic orbits, generalized Nehari manifold., generalized fountain theorems, Hamiltonian system, ground state.
Mathematics Subject Classification: Primary: 37J45; Secondary: 35B38, 70H0.
Citation: Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.  Google Scholar

[2]

T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, Elsevier B. V., Amsterdam, 2005.  Google Scholar

[3]

C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems, J. Math. Anal. Appl., 405 (2013), 438-452. doi: 10.1016/j.jmaa.2013.04.018.  Google Scholar

[4]

C. J. Batkam and F. Colin, On multiple solutions of a semilinear Schrödinger equation with periodic potential, Nonlinear Anal., 84 (2013), 39-49. doi: 10.1016/j.na.2013.02.006.  Google Scholar

[5]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.  Google Scholar

[6]

G. Chen and S. Ma, Homoclinic orbits of superlinear Hamiltonian systems, Proc. Amer. Math. Soc., 139 (2011), 3973-3983. doi: 10.1090/S0002-9939-2011-11185-7.  Google Scholar

[7]

V. Coti-Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 133-160. doi: 10.1007/BF01444526.  Google Scholar

[8]

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

[9]

Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38 (1999), 391-415. doi: 10.1016/S0362-546X(98)00204-1.  Google Scholar

[10]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.  Google Scholar

[11]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equat., 3 (1998), 441-472.  Google Scholar

[12]

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

[13]

A. Mao and S. Luan, Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems, Appl. Math. Comput., 214 (2009), 187-200. doi: 10.1016/j.amc.2009.03.084.  Google Scholar

[14]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. doi: 10.1007/s00032-005-0047-8.  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: 10.1007/BF02571356.  Google Scholar

[16]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. doi: 10.1007/BF02570817.  Google Scholar

[17]

M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var., 9 (2003), 601-619. doi: 10.1051/cocv:2003029.  Google Scholar

[18]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition, Springer-Verlag, Berlin, 1996.  Google Scholar

[19]

C. A. Stuart, Bifurcation into spectral gaps, Bull. Belg. Math. Soc., Supplement, (1995).  Google Scholar

[20]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[21]

A. Szulkin and T. Weth, The Method of Nehari Manifold, Int. Press, Somerville, MA, 2010.  Google Scholar

[22]

A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: 10.1006/jfan.2001.3798.  Google Scholar

[23]

K. Tanaka, Homoclinic orbits in a first order super-quadratic Hamiltonian system: Convergence of subharmonic orbits, J. Differential Equations, 94 (1991), 315-339. doi: 10.1016/0022-0396(91)90095-Q.  Google Scholar

[24]

J. Wang, J. X. Xu and F. B. Zhang, Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition, Discrete Contin. Dyn. Syst. Ser. A, 27 (2010), 1241-1257. doi: 10.3934/dcds.2010.27.1241.  Google Scholar

[25]

M. Willem, Minimax Theorems, Birkhauser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[26]

M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620-2627. doi: 10.1016/j.na.2009.11.009.  Google Scholar

[27]

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

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.  Google Scholar

[2]

T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, Elsevier B. V., Amsterdam, 2005.  Google Scholar

[3]

C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems, J. Math. Anal. Appl., 405 (2013), 438-452. doi: 10.1016/j.jmaa.2013.04.018.  Google Scholar

[4]

C. J. Batkam and F. Colin, On multiple solutions of a semilinear Schrödinger equation with periodic potential, Nonlinear Anal., 84 (2013), 39-49. doi: 10.1016/j.na.2013.02.006.  Google Scholar

[5]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.  Google Scholar

[6]

G. Chen and S. Ma, Homoclinic orbits of superlinear Hamiltonian systems, Proc. Amer. Math. Soc., 139 (2011), 3973-3983. doi: 10.1090/S0002-9939-2011-11185-7.  Google Scholar

[7]

V. Coti-Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 228 (1990), 133-160. doi: 10.1007/BF01444526.  Google Scholar

[8]

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

[9]

Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38 (1999), 391-415. doi: 10.1016/S0362-546X(98)00204-1.  Google Scholar

[10]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.  Google Scholar

[11]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equat., 3 (1998), 441-472.  Google Scholar

[12]

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

[13]

A. Mao and S. Luan, Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems, Appl. Math. Comput., 214 (2009), 187-200. doi: 10.1016/j.amc.2009.03.084.  Google Scholar

[14]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. doi: 10.1007/s00032-005-0047-8.  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: 10.1007/BF02571356.  Google Scholar

[16]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. doi: 10.1007/BF02570817.  Google Scholar

[17]

M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM Control Optim. Calc. Var., 9 (2003), 601-619. doi: 10.1051/cocv:2003029.  Google Scholar

[18]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition, Springer-Verlag, Berlin, 1996.  Google Scholar

[19]

C. A. Stuart, Bifurcation into spectral gaps, Bull. Belg. Math. Soc., Supplement, (1995).  Google Scholar

[20]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[21]

A. Szulkin and T. Weth, The Method of Nehari Manifold, Int. Press, Somerville, MA, 2010.  Google Scholar

[22]

A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: 10.1006/jfan.2001.3798.  Google Scholar

[23]

K. Tanaka, Homoclinic orbits in a first order super-quadratic Hamiltonian system: Convergence of subharmonic orbits, J. Differential Equations, 94 (1991), 315-339. doi: 10.1016/0022-0396(91)90095-Q.  Google Scholar

[24]

J. Wang, J. X. Xu and F. B. Zhang, Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition, Discrete Contin. Dyn. Syst. Ser. A, 27 (2010), 1241-1257. doi: 10.3934/dcds.2010.27.1241.  Google Scholar

[25]

M. Willem, Minimax Theorems, Birkhauser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[26]

M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Anal., 72 (2010), 2620-2627. doi: 10.1016/j.na.2009.11.009.  Google Scholar

[27]

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

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