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Homoclinic solutions for a class of second order nonautonomous singular Hamiltonian systems. (English) Zbl 07023152
Summary: We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems $$\ddot{u} + a \left(t\right) W_u \left(u\right) = 0$$, (HS) where $$- \infty < t < + \infty$$, $$u = \left(u_1, u_2, \ldots, u_N\right) \in \mathbb{R}^N \left(N \geq 3\right)$$, $$a : \mathbb{R} \rightarrow \mathbb{R}$$ is a continuous bounded function, and the potential $$W : \mathbb{R}^N \smallsetminus \{\xi \} \rightarrow \mathbb{R}$$ has a singularity at $$0 \neq \xi \in \mathbb{R}^N$$, and $$W_u \left(u\right)$$ is the gradient of $$W$$ at $$u$$. The novelty of this paper is that, for the case that $$N$$$$\geq 3$$ and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of $$W$$. Different from the cases that (HS) is autonomous $$\left(a \left(t\right) \equiv 1\right)$$ or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and $$N$$$$\geq 3$$. Besides the usual conditions on $$W$$, we need the assumption that $$a' \left(t\right) < 0$$ for all $$t \in \mathbb{R}$$ to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.
MSC:
 34-XX Ordinary differential equations 35-XX Partial differential equations
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 [1] Ambrosetti, A.; Zelati, V. C., Periodic Solutions of Singular Lagrangian Systems. Periodic Solutions of Singular Lagrangian Systems, Progress in Nonlinear Differential Equations and Their Applications, 10, (1993), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 0785.34032 [2] Gordon, W. B., Conservative dynamical systems involving strong force, Transactions of the American Mathematical Society, 204, 113-115, (1975) · Zbl 0276.58005 [3] Rabinowitz, P. H., Periodic solutions of Hamiltonian systems, Communications on Pure and Applied Mathematics, 31, 2, 157-184, (1978) · Zbl 0358.70014 [4] Tanaka, K., Homoclinic orbits for a singular second order Hamiltonian system, Annales de l’Institut Henri Poincaré C, 7, 427-438, (1990) · Zbl 0712.58026 [5] Bahri, A.; Rabinowitz, P. H., A minimax method for a class of Hamiltonian systems with singular potentials, Journal of Functional Analysis, 82, 2, 412-428, (1989) · Zbl 0681.70018 [6] Caldiroli, P., Existence and multiplicity of homoclinic orbits for potentials on unbounded domains, Proceedings of the Royal Society of Edinburgh A, 124, 2, 317-339, (1994) · Zbl 0807.34058 [7] Bessi, U., Multiple homoclinic orbits for autonomous singular potentials, Proceedings of the Royal Society of Edinburgh A, 124, 4, 785-802, (1994) · Zbl 0812.58088 [8] Janczewska, J.; Maksymiuk, J., Homoclinic orbits for a class of singular second order Hamiltonian systems in $$\mathbb{R}^3$$, Central European Journal of Mathematics, 10, 6, 1920-1927, (2012) · Zbl 1261.34037 [9] Bertotti, M. L.; Jeanjean, L., Multiplicity of homoclinic solutions for singular second-order conservative systems, Proceedings of the Royal Society of Edinburgh A, 126, 6, 1169-1180, (1996) · Zbl 0868.34001 [10] Caldiroli, P.; de Coster, C., Multiple homoclinics for a class of singular Hamiltonian systems, Journal of Mathematical Analysis and Applications, 211, 2, 556-573, (1997) · Zbl 0887.58017 [11] Rabinowitz, P. H., Homoclinics for a singular Hamiltonian system, Geometric Analysis and the Calculus of Variations, 267-296, (1996), Cambridge, Mass, USA: International Press, Cambridge, Mass, USA · Zbl 0936.37035 [12] Caldiroli, P.; Jeanjean, L., Homoclinics and heteroclinics for a class of conservative singular hamiltonian systems, Journal of Differential Equations, 136, 1, 76-114, (1997) · Zbl 0887.34044 [13] Borges, M. J., Heteroclinic and homoclinic solutions for a singular Hamiltonian system, European Journal of Applied Mathematics, 17, 1, 1-32, (2006) · Zbl 1160.37390 [14] Izydorek, M.; Janczewska, J., The shadowing chain lemma for singular Hamiltonian systems involving strong forces, Central European Journal of Mathematics, 10, 6, 1928-1939, (2012) · Zbl 1269.37015 [15] Séré, E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Mathematische Zeitschrift, 209, 1, 27-42, (1992) · Zbl 0725.58017 [16] Sere, E., Looking for the Bernoulli shift, Annales de l’Institut Henri Poincaré C, 10, 561-590, (1993) · Zbl 0803.58013 [17] Caldiroli, P.; Montecchiari, P., Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Communications on Applied Nonlinear Analysis, 1, 97-129, (1994) · Zbl 0867.70012 [18] Zelati, V. C.; Rabinowitz, P. H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, Journal of the American Mathematical Society, 4, 4, 693-727, (1991) · Zbl 0744.34045 [19] Rabinowitz, P. H., Multibump solutions for an almost periodically forced singular Hamiltonian system, Electronic Journal of Differential Equations, 1995, 12, 1-21, (1995) · Zbl 0828.34034 [20] Izydorek, M.; Janczewska, J., Connecting orbits for a periodically forced singular planar Newtonian system, Journal of Fixed Point Theory and Applications, 12, 1-2, 59-67, (2012) · Zbl 1271.34051 [21] Costa, D. G.; Tehrani, H., On a class of singular second-order Hamiltonian systems with infinitely many homoclinic solutions, Journal of Mathematical Analysis and Applications, 412, 1, 200-211, (2014) · Zbl 1317.34092 [22] Greco, C., Periodic solutions of a class of singular Hamiltonian systems, Nonlinear Analysis, 12, 3, 259-269, (1988) · Zbl 0648.34048 [23] Rabinowitz, P. H., Periodic and heteroclinic orbits for a periodic Hamiltonian system, Annales de l’Institut Henri Poincaré C, 6, 331-346, (1989) · Zbl 0701.58023
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