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On a degree associated with the Gross-Pitaevskii system with a large parameter
Subharmonic solutions for a class of Lagrangian systems
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 |
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 [
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. |
[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. |
[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. |
[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. |
[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.
|
[6] |
K. Gęba, 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.
doi: 10.1006/jdeq.2000.3818. |
[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. |
[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. |
[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. |
[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. |
[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.
doi: 10.1007/978-1-4757-2061-7. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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.
|
[6] |
K. Gęba, 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.
doi: 10.1006/jdeq.2000.3818. |
[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. |
[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. |
[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. |
[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. |
[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.
doi: 10.1007/978-1-4757-2061-7. |
[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. |
[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. |
[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. |
[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. |
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