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Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. (English) Zbl 0919.34046
The paper deals with the existence of homoclinic orbits for second-order time-dependent Hamiltonian systems of the type $\ddot{q} - L(t) q + W_q(t,q) = 0, \qquad q \in \mathbb{R} ^n.$ In nontechnical terms, $$W$$ is assumed to be superquadratic at infinity and subquadratic at the origin. Also, $$L(t)$$ tends to a multiple of the identity matrix as $$| t| \to \infty$$. Under such conditions, the existence of a homoclinic orbit is proven. The authors apply variational methods, such as the mountain pass theorem. The result generalizes previous results, such as work of P. H. Rabinowitz and K. Tanaka [Math. Z. 206, No. 3, 473-499 (1991; Zbl 0707.58022)]. Other generalizations can be found in work of S. Wu and O. H. Miyagaki.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems, general theory 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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##### References:
 [1] Ambrosetti, A.; Bertotti, M.L., Homoclinic for second order conservative systems in P.D.E and related subjects, Pitman research notes in mathematical science, (1992), p. 21-37 · Zbl 0804.34046 [2] G. Arioli, A. Szulkin, Homoclinic solutions for a class of systems of second order differential equations, 5, Dept. of Math, Univ. Stockholm, Sweden, 1995 · Zbl 0857.34048 [3] Costa, D.G., On a class of elliptic systems in $$R$$^N, Elect. J. differential equations, 1994, 1-14, (1994) [4] Coti-Zelati, V.; Ekeland, I.; Séré, E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. ann., 288, 133-160, (1990) · Zbl 0731.34050 [5] Coti-Zelati, V.; Rabinowitz, P.H., Homoclinic orbits for a second order Hamiltonian system possessing superquadratic potentials, J. amer. math. soc., 4, 693-727, (1992) · Zbl 0744.34045 [6] Ding, Y., Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear anal. TMA, 25, 1095-1113, (1995) · Zbl 0840.34044 [7] Ding, Y.; Ni, W.Y., On the existence of positive entire solutions of a semilinear elliptic equations, Arch. rat. mech. anal., 91, 283-308, (1986) · Zbl 0616.35029 [8] Felmer, P.; Silva, E.A.B., Homoclinic and periodic orbits for Hamiltonian systems, Relatório de pesquisa unicamp, 5, (1996) [9] Jianfu, Y.; Xiping, Z., On the existence of nontrivial solution of quasilinear elliptic boundary value problem for unbounded domains, Acta math. sci., 7, 341-359, (1987) · Zbl 0674.35030 [10] Korman, P.; Lazer, A.C., Homoclinic orbits for a class of symmetric Hamiltonian systems, Elect. J. differential equations, 1994, 1-10, (1994) [11] Lions, P.L., The concentration-compactness principle in the calculus of variations, the locally compact case, part 1, Ann. I.H.P. anal. non linéaire, 1, 109-145, (1984) · Zbl 0541.49009 [12] Lions, P.L., The concentration-compactness principle in the calculus of variations, the locally compact case, part 2, Ann. I.H.P. anal. non linéaire, 1, 223-283, (1984) · Zbl 0704.49004 [13] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag Berlin · Zbl 0676.58017 [14] Omana, W.; Willem, M., Homoclinic orbits for a class of Hamiltonian systems, Differential integral equations, 5, 1115-1120, (1992) · Zbl 0759.58018 [15] Rabinowitz, P.H., Periodic and heteroclinic orbits for a periodic Hamiltonian systems, Ann. I.H.P. anal. non linéaire, 6, 331-346, (1989) · Zbl 0701.58023 [16] Rabinowitz, P.H., Homoclinic orbits for a class of Hamiltonian systems, Proc. roy. soc. edinburg, 114A, 33-38, (1990) · Zbl 0705.34054 [17] Rabinowitz, P.H., Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. variations, 1, 1-36, (1993) · Zbl 0791.34042 [18] Rabinowitz, P.H.; Tanaka, K., Some results on connecting orbits for a class of Hamiltonian systems, Math Z., 206, 473-499, (1991) · Zbl 0707.58022 [19] Séré, E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math Z., 209, 27-42, (1992) · Zbl 0725.58017 [20] Serra, E.; Tarallo, M.; Terracini, S., On the existence of homoclinic solutions for almost periodic second order systems, Ann. I.H.P. anal. non linéaire, 13, 783-812, (1996) · Zbl 0873.58032
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