Two homoclinic orbits for some second-order Hamiltonian systems
Keywords
Indefinite problem, concave-convex, variational methodsAbstract
This paper is concerned with the existence of homoclinic orbits for a class of second order Hamiltonian systems considering a non-periodic potential and a weaker Ambrosetti-Rabinowitz condition. By considering an auxiliary problem, we show the existence of two different approximative sequences of periodic solutions, the first one of mountain pass type and the second one of local minima. We obtain two different homoclinic orbits by passing to the limit in such sequences. As a relevant application, we obtain another homoclinic solution for the Hamiltonian system studied in \cite{IJ}.References
A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova 89 (1993), 177–194.
A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
V. Coti Zelati and P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potenials, J. Amer. Math. Soc. 4 (1991), 693–727.
P.L. Felmer, Variational methods in Hamiltonian systems, Dynamical Systems (Temuco, 1991/1992), Travaux en Cours, vol. 52, Hermann, Paris, 1996, 151–178.
M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations 219 (2005), 375–389.
M. Izydorek and J. Janczewska, Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl. 335, 1119–1127.
H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier–Villars, Pairs, 1897–1899.
P.H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 33–38.
L.P. Shil’nikov, Homoclinic trajectories: From Poincaré to the present, Mathematical Events of the Twentieth Century, Springer, Berlin, 2006, 347–370.
Ye, Yiwei and Tang, Chun-Lei Multiple homoclinic solutions for second-order perturbed Hamiltonian systems, Stud. Appl. Math. 132 (2014), no. 2, 112–137.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
