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Subharmonic solutions for a class of Lagrangian systems

Abstract

We prove that second order Hamiltonian systems with a potential of class C1, 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 [P. H. Rabinowitz, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38]. Indeed, we weaken the latter condition in a neighbourhood of the origin. We will also discuss when subharmonics pass to a nontrivial homoclinic orbit.

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Category:
Articles
Type:
artykuły w czasopismach
Published in:
Discrete and Continuous Dynamical Systems-Series S no. 12, pages 1841 - 1850,
ISSN: 1937-1632
Language:
English
Publication year:
2019
Bibliographic description:
Bahrouni A., Izydorek M., Janczewska J.: Subharmonic solutions for a class of Lagrangian systems// Discrete and Continuous Dynamical Systems-Series S -Vol. 12,iss. 7 (2019), s.1841-1850
DOI:
Digital Object Identifier (open in new tab) 10.3934/dcdss.2019121
Bibliography: test
  1. A. Abbondandolo, Morse Theory for Hamiltonian Systems, Chapman and Hall/CRC Re- search Notes in Mathematics 425, Chapman and Hall/CRC, Boca Raton, FL, 2001.
  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. open in new tab
  3. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. open in new tab
  4. K. Ch. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser Boston, Inc., Boston, MA, 1993. open in new tab
  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. open in new tab
  6. K. Gȩba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications, Studia Math., 134 (1999), 217-233.
  7. M. Izydorek, A cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems, J. Differential Equations, 170 (2001), 22-50. open in new tab
  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. open in new tab
  9. M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamil- tonian systems, J. Differential Equations, 219 (2005), 375-389. open in new tab
  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. open in new tab
  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. open in new tab
  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.
  13. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci. 74, Springer-Verlag, New York, 1989. open in new tab
  14. P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38. open in new tab
  15. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differ- ential Equations, CBMS Regional Conference Series in Mathematics 65, Amer. Math. Soc., Providence, RI, 1986. open in new tab
  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. open in new tab
  17. K. Tanaka, Homoclinic orbits for a singular second order Hamiltonian system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 427-438. open in new tab
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Gdańsk University of Technology

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