# American Institute of Mathematical Sciences

• Previous Article
A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity
• DCDS Home
• This Issue
• Next Article
Density of fiberwise orbits in minimal iterated function systems on the circle
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

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.
Citation: Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete and 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. [2] T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, Elsevier B. V., Amsterdam, 2005. [3] C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems, J. Math. Anal. Appl., 405 (2013), 438-452. 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), 39-49. 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), 486-490. doi: 10.2307/2044999. [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. [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. [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. [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. [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. [11] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equat., 3 (1998), 441-472. [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. [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. [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. [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. [16] E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. 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), 601-619. doi: 10.1051/cocv:2003029. [18] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition, Springer-Verlag, Berlin, 1996. [19] C. A. Stuart, Bifurcation into spectral gaps, Bull. Belg. Math. Soc., Supplement, (1995). [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. [21] A. Szulkin and T. Weth, The Method of Nehari Manifold, Int. Press, Somerville, MA, 2010. [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. [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. [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. [25] M. Willem, Minimax Theorems, Birkhauser, Boston, 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), 2620-2627. doi: 10.1016/j.na.2009.11.009. [27] W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358. doi: 10.1007/s002290170032.

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, Elsevier B. V., Amsterdam, 2005. [3] C. J. Batkam and F. Colin, Generalized fountain theorem and applications to strongly indefinite semilinear problems, J. Math. Anal. Appl., 405 (2013), 438-452. 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), 39-49. 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), 486-490. doi: 10.2307/2044999. [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. [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. [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. [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. [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. [11] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equat., 3 (1998), 441-472. [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. [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. [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. [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. [16] E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. 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), 601-619. doi: 10.1051/cocv:2003029. [18] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Second Edition, Springer-Verlag, Berlin, 1996. [19] C. A. Stuart, Bifurcation into spectral gaps, Bull. Belg. Math. Soc., Supplement, (1995). [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. [21] A. Szulkin and T. Weth, The Method of Nehari Manifold, Int. Press, Somerville, MA, 2010. [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. [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. [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. [25] M. Willem, Minimax Theorems, Birkhauser, Boston, 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), 2620-2627. doi: 10.1016/j.na.2009.11.009. [27] W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358. doi: 10.1007/s002290170032.
 [1] Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210 [2] Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139 [3] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 [4] A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419 [5] Fatima Ezzahra Lembarki, Jaume Llibre. Periodic orbits for a generalized Friedmann-Robertson-Walker Hamiltonian system in dimension $6$. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1165-1211. doi: 10.3934/dcdss.2015.8.1165 [6] Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257 [7] Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195 [8] Jian Zhang, Wen Zhang. Existence and decay property of ground state solutions for Hamiltonian elliptic system. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2433-2455. doi: 10.3934/cpaa.2019110 [9] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128 [10] Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645 [11] Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214 [12] Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778 [13] Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929 [14] Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 [15] Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Communications on Pure and Applied Analysis, 2019, 18 (1) : 425-434. doi: 10.3934/cpaa.2019021 [16] Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure and Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269 [17] Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114 [18] Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967 [19] Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285 [20] Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039

2020 Impact Factor: 1.392