American Institute of Mathematical Sciences

Advanced
November  2019, 12(7): 1841-1850. doi: 10.3934/dcdss.2019121

Subharmonic solutions for a class of Lagrangian systems

Anouar Bahrouni 1, , Marek Izydorek 2,, and  Joanna Janczewska 2,

1. 

Department of Mathematics, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia
2. 

Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland

* Corresponding author: Marek Izydorek

Received  April 2018 Revised  May 2018 Published  December 2018

Fund Project: M. Izydorek and J. Janczewska are supported by Grant BEETHOVEN2 of the National Science Centre, Poland, no. 2016/23/G/ST1/04081.

We prove that second order Hamiltonian systems $ -\ddot{u} = V_{u}(t,u) $ with a potential $ V\colon \mathbb{R} \times \mathbb{R} ^N\to \mathbb{R} $ of class $ C^1 $, periodic in time and superquadratic at infinity with respect to the space variable have subharmonic solutions. Our intention is to generalise a result on subharmonics for Hamiltonian systems with a potential satisfying the global Ambrosetti-Rabinowitz condition from [14]. Indeed, we weaken the latter condition in a neighbourhood of $ 0\in \mathbb{R} ^N $. We will also discuss when subharmonics pass to a nontrivial homoclinic orbit.

Keywords: Periodic solution, Lagrangian system, critical point, homoclinic orbit.
Mathematics Subject Classification: Primary: 37J45, 70H03; Secondary: 34C25, 34C37.
Citation: Anouar Bahrouni, Marek Izydorek, Joanna Janczewska. Subharmonic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1841-1850. doi: 10.3934/dcdss.2019121
References:
[1]

A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman and Hall/CRC Research Notes in Mathematics 425, Chapman and Hall/CRC, Boca Raton, FL, 2001.  Google Scholar

[2]

A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems, Progr. Nonlinear Differential Equations Appl. 10, Birkh ser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0319-3.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[4]

K. Ch. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[5]

J. Ciesielski, J. Janczewska and N. Waterstraat, On the existence of homoclinic type solutions of inhomogenous Lagrangian systems, Differential and Integral Equations, 30 (2017), 259-272.   Google Scholar

[6]

K. Gęba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications, Studia Math., 134 (1999), 217-233.   Google Scholar

[7]

M. Izydorek, A cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems, J. Differential Equations, 170 (2001), 22-50.  doi: 10.1006/jdeq.2000.3818.  Google Scholar

[8]

M. Izydorek, Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems, Nonl. Analysis Ser. A: Theory Methods, 51 (2002), 33-66.  doi: 10.1016/S0362-546X(01)00811-2.  Google Scholar

[9]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389.  doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[10]

M. Izydorek and J. Janczewska, The shadowing chain lemma for singular Hamiltonian systems involving strong forces, Cent. Eur. J. Math., 10 (2012), 1928-1939.  doi: 10.2478/s11533-012-0107-6.  Google Scholar

[11]

J. Janczewska, An approximative scheme of finding almost homoclinic solutions for a class of Newtonian systems, Topol. Methods Nonlinear Anal., 33 (2009), 169-177.  doi: 10.12775/TMNA.2009.012.  Google Scholar

[12]

J. Janczewska, Homoclinic solutions for a class of autonomous second order Hamiltonian systems with a superquadratic potential, Topol. Methods Nonlinear Anal., 36 (2010), 19-26.   Google Scholar

[13]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci. 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[14]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  doi: 10.1017/S0308210500024240.  Google Scholar

[15]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, Amer. Math. Soc., Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[16]

E. Serra, M. Tarallo and S. Terracini, On the existence of homoclinic solutions for almost periodic second order systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 783-812. doi: 10.1016/S0294-1449(16)30123-8.  Google Scholar

[17]

K. Tanaka, Homoclinic orbits for a singular second order Hamiltonian system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 427-438. doi: 10.1016/S0294-1449(16)30285-2.  Google Scholar

show all references

References:
[1]

A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman and Hall/CRC Research Notes in Mathematics 425, Chapman and Hall/CRC, Boca Raton, FL, 2001.  Google Scholar

[2]

A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems, Progr. Nonlinear Differential Equations Appl. 10, Birkh ser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0319-3.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[4]

K. Ch. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[5]

J. Ciesielski, J. Janczewska and N. Waterstraat, On the existence of homoclinic type solutions of inhomogenous Lagrangian systems, Differential and Integral Equations, 30 (2017), 259-272.   Google Scholar

[6]

K. Gęba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications, Studia Math., 134 (1999), 217-233.   Google Scholar

[7]

M. Izydorek, A cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems, J. Differential Equations, 170 (2001), 22-50.  doi: 10.1006/jdeq.2000.3818.  Google Scholar

[8]

M. Izydorek, Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems, Nonl. Analysis Ser. A: Theory Methods, 51 (2002), 33-66.  doi: 10.1016/S0362-546X(01)00811-2.  Google Scholar

[9]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219 (2005), 375-389.  doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[10]

M. Izydorek and J. Janczewska, The shadowing chain lemma for singular Hamiltonian systems involving strong forces, Cent. Eur. J. Math., 10 (2012), 1928-1939.  doi: 10.2478/s11533-012-0107-6.  Google Scholar

[11]

J. Janczewska, An approximative scheme of finding almost homoclinic solutions for a class of Newtonian systems, Topol. Methods Nonlinear Anal., 33 (2009), 169-177.  doi: 10.12775/TMNA.2009.012.  Google Scholar

[12]

J. Janczewska, Homoclinic solutions for a class of autonomous second order Hamiltonian systems with a superquadratic potential, Topol. Methods Nonlinear Anal., 36 (2010), 19-26.   Google Scholar

[13]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci. 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[14]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  doi: 10.1017/S0308210500024240.  Google Scholar

[15]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, Amer. Math. Soc., Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[16]

E. Serra, M. Tarallo and S. Terracini, On the existence of homoclinic solutions for almost periodic second order systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 783-812. doi: 10.1016/S0294-1449(16)30123-8.  Google Scholar

[17]

K. Tanaka, Homoclinic orbits for a singular second order Hamiltonian system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 427-438. doi: 10.1016/S0294-1449(16)30285-2.  Google Scholar

[1]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[2]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
[3]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
[4]

Zhihong Xia. Homoclinic points and intersections of Lagrangian submanifold. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 243-253. doi: 10.3934/dcds.2000.6.243
[5]

Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021
[6]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129
[7]

Zhirong He, Weinian Zhang. Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 841-854. doi: 10.3934/dcds.2009.24.841
[8]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883
[9]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[10]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[11]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[12]

Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385
[13]

Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218
[14]

Boris Buffoni, Laurent Landry. Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 75-116. doi: 10.3934/dcds.2010.27.75
[15]

Wenhua Qiu, Jianguo Si. On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point. Communications on Pure & Applied Analysis, 2015, 14 (2) : 421-437. doi: 10.3934/cpaa.2015.14.421
[16]

Gautier Picot. Energy-minimal transfers in the vicinity of the lagrangian point $L_1$. Conference Publications, 2011, 2011 (Special) : 1196-1205. doi: 10.3934/proc.2011.2011.1196
[17]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[18]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911
[19]

Samir Adly, Daniel Goeleven, Dumitru Motreanu. Periodic and homoclinic solutions for a class of unilateral problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 579-590. doi: 10.3934/dcds.1997.3.579
[20]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483

2019 Impact Factor: 1.233

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences