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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

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