Subharmonic solutions for a class of Lagrangian systems

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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

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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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