Homoclinic orbits of first-order superquadratic Hamiltonian systems

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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

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

Received May 2013 Revised November 2013 Published March 2014

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

References:

[1]	R. A. Adams, Sobolev Spaces,, Academic Press, (1975).
[2]	T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods,, Elsevier B. V., (2005).
[3]	C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems,, J. Math. Anal. Appl., 405 (2013), 438. doi: 10.1016/j.jmaa.2013.04.018.
[4]	C. J. Batkam and F. Colin, On multiple solutions of a semilinear Schrödinger equation with periodic potential,, Nonlinear Anal., 84 (2013), 39. doi: 10.1016/j.na.2013.02.006.
[5]	H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.2307/2044999.
[6]	G. Chen and S. Ma, Homoclinic orbits of superlinear Hamiltonian systems,, Proc. Amer. Math. Soc., 139 (2011), 3973. doi: 10.1090/S0002-9939-2011-11185-7.
[7]	V. Coti-Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems,, Math. Ann., 228 (1990), 133. doi: 10.1007/BF01444526.
[8]	Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms,, Commun. Contemp. Math., 8 (2006), 453. doi: 10.1142/S0219199706002192.
[9]	Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry,, Nonlinear Anal., 38 (1999), 391. doi: 10.1016/S0362-546X(98)00204-1.
[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. doi: 10.1017/S0308210500013147.
[11]	W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation,, Adv. Differ. Equat., 3 (1998), 441.
[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.
[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. doi: 10.1016/j.amc.2009.03.084.
[14]	A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals,, Milan J. Math., 73 (2005), 259. doi: 10.1007/s00032-005-0047-8.
[15]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 473. doi: 10.1007/BF02571356.
[16]	E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Math. Z., 209 (1992), 27. doi: 10.1007/BF02570817.
[17]	M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent,, ESAIM Control Optim. Calc. Var., 9 (2003), 601. doi: 10.1051/cocv:2003029.
[18]	M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Second Edition, (1996).
[19]	C. A. Stuart, Bifurcation into spectral gaps,, Bull. Belg. Math. Soc., (1995).
[20]	A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems,, J. Funct. Anal., 257 (2009), 3802. doi: 10.1016/j.jfa.2009.09.013.
[21]	A. Szulkin and T. Weth, The Method of Nehari Manifold,, Int. Press, (2010).
[22]	A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems,, J. Funct. Anal., 187 (2001), 25. doi: 10.1006/jfan.2001.3798.
[23]	K. Tanaka, Homoclinic orbits in a first order super-quadratic Hamiltonian system: Convergence of subharmonic orbits,, J. Differential Equations, 94 (1991), 315. doi: 10.1016/0022-0396(91)90095-Q.
[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. doi: 10.3934/dcds.2010.27.1241.
[25]	M. Willem, Minimax Theorems,, Birkhauser, (1996). doi: 10.1007/978-1-4612-4146-1.
[26]	M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities,, Nonlinear Anal., 72 (2010), 2620. doi: 10.1016/j.na.2009.11.009.
[27]	W. Zou, Variant fountain theorems and their applications,, Manuscripta Math., 104 (2001), 343. doi: 10.1007/s002290170032.

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References:

[1]	R. A. Adams, Sobolev Spaces,, Academic Press, (1975).
[2]	T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods,, Elsevier B. V., (2005).
[3]	C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems,, J. Math. Anal. Appl., 405 (2013), 438. doi: 10.1016/j.jmaa.2013.04.018.
[4]	C. J. Batkam and F. Colin, On multiple solutions of a semilinear Schrödinger equation with periodic potential,, Nonlinear Anal., 84 (2013), 39. doi: 10.1016/j.na.2013.02.006.
[5]	H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.2307/2044999.
[6]	G. Chen and S. Ma, Homoclinic orbits of superlinear Hamiltonian systems,, Proc. Amer. Math. Soc., 139 (2011), 3973. doi: 10.1090/S0002-9939-2011-11185-7.
[7]	V. Coti-Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems,, Math. Ann., 228 (1990), 133. doi: 10.1007/BF01444526.
[8]	Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms,, Commun. Contemp. Math., 8 (2006), 453. doi: 10.1142/S0219199706002192.
[9]	Y. H. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry,, Nonlinear Anal., 38 (1999), 391. doi: 10.1016/S0362-546X(98)00204-1.
[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. doi: 10.1017/S0308210500013147.
[11]	W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation,, Adv. Differ. Equat., 3 (1998), 441.
[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.
[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. doi: 10.1016/j.amc.2009.03.084.
[14]	A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals,, Milan J. Math., 73 (2005), 259. doi: 10.1007/s00032-005-0047-8.
[15]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 473. doi: 10.1007/BF02571356.
[16]	E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Math. Z., 209 (1992), 27. doi: 10.1007/BF02570817.
[17]	M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent,, ESAIM Control Optim. Calc. Var., 9 (2003), 601. doi: 10.1051/cocv:2003029.
[18]	M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Second Edition, (1996).
[19]	C. A. Stuart, Bifurcation into spectral gaps,, Bull. Belg. Math. Soc., (1995).
[20]	A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems,, J. Funct. Anal., 257 (2009), 3802. doi: 10.1016/j.jfa.2009.09.013.
[21]	A. Szulkin and T. Weth, The Method of Nehari Manifold,, Int. Press, (2010).
[22]	A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems,, J. Funct. Anal., 187 (2001), 25. doi: 10.1006/jfan.2001.3798.
[23]	K. Tanaka, Homoclinic orbits in a first order super-quadratic Hamiltonian system: Convergence of subharmonic orbits,, J. Differential Equations, 94 (1991), 315. doi: 10.1016/0022-0396(91)90095-Q.
[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. doi: 10.3934/dcds.2010.27.1241.
[25]	M. Willem, Minimax Theorems,, Birkhauser, (1996). doi: 10.1007/978-1-4612-4146-1.
[26]	M. Yang, Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities,, Nonlinear Anal., 72 (2010), 2620. doi: 10.1016/j.na.2009.11.009.
[27]	W. Zou, Variant fountain theorems and their applications,, Manuscripta Math., 104 (2001), 343. doi: 10.1007/s002290170032.

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