## Infinitely many homoclinic solutions for a class of indefinite perturbed second-order Hamiltonian systems.(English)Zbl 1367.34052

The existence of homoclinic orbits is studied for the class of differential equations $-\ddot{u}(t)+L(t)u(t)=W_u(t,u(t))+G_u(t,u(t)).$ The existence of infinitely many homoclinic solutions is proven by using the theory of Bolle’s perturbation method in critical point. The paper reports some generalizations of known results.

### MSC:

 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 34E10 Perturbations, asymptotics of solutions to ordinary differential equations 34C14 Symmetries, invariants of ordinary differential equations
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### References:

 [1] Bahri, A.; Berestycki, H., A perturbation method in critical point theory and applications, Trans. Am. Math. Soc., 267, 1-32, (1981) · Zbl 0476.35030 [2] Berezin F.A., Shubin M.A.: The Schrödinger Equation. Kluwer Academic, London (1991) · Zbl 0749.35001 [3] Bolle, P., On the Bolza problem, J. Differ. Equ., 152, 274-288, (1999) · Zbl 0923.34025 [4] Bolle, P.; Ghoussoub, N.; Tehari, H., The multiplicity of solutions in nonhomogenous boundary value problems, Manuscr. Math., 101, 325-350, (2002) · Zbl 0963.35001 [5] Candela, A.M.; Palmieri, G.; Salvatore, A., Radial solutions of semilinear elliptic equations with broken symmetry, Topol. Methods Nonlinear Anal., 27, 117-132, (2006) · Zbl 1135.35339 [6] Ding, Y.H.; Girardi, M., Periodic and homoclinic solutions to a class of Hamilton systems with potential changing sign, Dyn. Syst. Appl., 2, 131-145, (1993) · Zbl 0771.34031 [7] Ding, Y.H.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25, 1095-1113 (1995) · Zbl 0840.34044 [8] Izydorek, M.; Janczewska, J., Homoclinic solutions for a class of second order Hamilton systems, J. Differ. Equ., 219, 375-389, (2005) · Zbl 1080.37067 [9] Korman, P., Lazer, A.C.: Homoclinic orbits for a class of symmetric Hamiltonian systems. Electron. J. Differ. Equ. 1994, 1-10 (1994) · Zbl 0788.34042 [10] Omana, W.; Willem, M., Homoclinic orbits for a class of Hamiltonian systems, Differ. Integral Equ., 5, 1115-1120, (1992) · Zbl 0759.58018 [11] Ou, Z.Q.; Tang, C.L., Existence of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 291, 203-213, (2004) · Zbl 1057.34038 [12] Rabinowitz, P.H., Multiple critical points of perturbed symmetric functionals, Trans. Am. Math. Soc., 272, 753-769, (1982) · Zbl 0589.35004 [13] Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence (1986) · Zbl 0609.58002 [14] Rabinowitz, P.H., Homoclinic orbits for a class of Hamilton systems, Proc. R. Soc. Ediburgh Sect. A, 114, 33-38, (1990) · Zbl 0705.34054 [15] 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 [16] Salvatore, A.: Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Dynamical systems and differential equations (Wilmington, NC, 2002). Discrete Contin. Dyn. Syst. Suppl. 778-787 (2003) · Zbl 1073.34046 [17] Salvatore, A., Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Stud., 3, 1-23, (2003) · Zbl 1221.35400 [18] Schechter, M.; Zou, W., Infinitely many solutions to perturbed elliptic equations, J. Funct. Anal., 228, 1-38, (2005) · Zbl 1139.35346 [19] Struwe, M., Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscr. Math., 32, 335-364, (1980) · Zbl 0456.35031 [20] Tang, X.H.; Xiao, L., Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl., 351, 586-594, (2009) · Zbl 1153.37408 [21] Tang, X.H.; Lin, X.Y., Homoclinic solutions for a class of second-order Hamiltonian systems, J. Math. Anal. Appl., 354, 539-549, (2009) · Zbl 1179.37082 [22] Tehrani, H.T., Infinitely many solutions for indefinite semilinear elliptic equations without symmetry, Commun. Partial Differ. Equ., 21, 541-557, (1996) · Zbl 0855.35044 [23] Wan, L.L.; Tang, C.L., Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Discrete Contin. Dyn. Syst. Ser. B, 15, 255-271, (2011) · Zbl 1216.34033 [24] Zelati, V.C.; Rabinowitz, P.H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Am. Math. Soc, 4, 693-727, (1991) · Zbl 0744.34045
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