Opuscula Math. 40, no. 1 (2020), 21-36
https://doi.org/10.7494/OpMath.2020.40.1.21

Opuscula Mathematica

# Some multiplicity results of homoclinic solutions for second order Hamiltonian systems

Sara Barile

Abstract. We look for homoclinic solutions $$q:\mathbb{R} \rightarrow \mathbb{R}^N$$ to the class of second order Hamiltonian systems $-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}$ where $$L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}$$ and $$a,b: \mathbb{R}\rightarrow \mathbb{R}$$ are positive bounded functions, $$G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}$$ are positive homogeneous functions and $$f:\mathbb{R}\rightarrow\mathbb{R}^N$$. Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if $$f\equiv 0$$ and the existence of at least three solutions if $$f$$ is not trivial but small enough.

Keywords: second order Hamiltonian systems, homoclinic solutions, variational methods, compact embeddings.

Mathematics Subject Classification: 34C37, 58E05, 70H05.

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1. R.A. Adams, C. Essex, Calculus. A Complete Course, 7th ed., Pearson Canada, Toronto, 2010.
2. J. Ciesielski, J. Janczewska, N. Waterstraat, On the existence of homoclinic type solutions of inhomogenous Lagrangian systems, Differential Integral Equations 30 (2017), 259-272.
3. V. Coti Zelati, P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 (1991) 4, 693-727.
4. Y.H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal. 25 (1995) 11, 1095-1113.
5. A. Fonda, M. Garrione, P. Gidoni, Periodic perturbations of Hamiltonian systems, Adv. Nonlinear Anal. 5 (2016) 4, 367-382.
6. A. Fonda, R. Toader, Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth, Adv. Nonlinear Anal. 8 (2019) 1, 583-602.
7. M. Izydorek, J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations 219 (2005), 375-389.
8. L. Ljusternik, L. Schnirelmann, Méthodes Topologiques dans les Problèmes Variationnels, Hermann, Paris, 1934.
9. W. Omana, M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations 5 (1992) 5, 1115-1120.
10. S.I. Pohozaev, The fibering method in nonlinear variational problems, Pitman Res. Notes Math. Ser. 365 (1997), 35-88.
11. S.I. Pohozaev, The fibering method and its applications to nonlinear boundary value problems, Rend. Istit. Mat. Univ. Trieste XXXI (1999), 235-305.
12. H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris, 1899.
13. P.H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 1-2, 33-38.
14. P.H. Rabinowitz, K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z. 206 (1991) 3, 473-499.
15. A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Proceedings of the Second World Congress of Nonlinear Analysis, Part 8 (Athens, 1996), Nonlinear Anal. 30 (1997) 8, 4849-4857.
16. A. Salvatore, Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Dynamical systems and differential equations (Wilmington, NC, 2002), Discrete Contin. Dyn. Syst. 2003, suppl., 778-787.
17. K. Tanaka, Homoclinic orbits for a singular second order Hamiltonian system, Ann. Inst. H. Poincaré 7 (1990) 5, 427-438.
18. X.H. Tang, X. Lin, Homoclinic solutions for a class of second-order Hamiltonian systems, J. Math. Anal. Appl. 354 (2009), 539-549.
19. L.L. Wan, C.L. Tang, Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Discrete Contin. Dyn. Syst. 15 (2011), 255-271.
20. W. Zou, S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett. 16 (2003), 1283-1287.
• Communicated by Vicentiu D. Radulescu.
• Revised: 2019-03-26.
• Accepted: 201903-26.
• Published online: 2020-02-17.