Taiwanese Journal of Mathematics

HOMOCLINIC SOLUTIONS FOR SUBQUADRATIC HAMILTONIAN SYSTEMS WITHOUT COERCIVE CONDITIONS

Ziheng Zhang, Tian Xiang, and Rong Yuan

Full-text: Open access

Abstract

In this paper we investigate the existence and multiplicity of classical homoclinic solutions for the following second order Hamiltonian systems $$ (\mbox{HS}) \ddot u-L(t)u+\nabla W(t,u)=0, $$ where $L\in C(\mathbb R,\mathbb R^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb R$, $W\in C^1(\mathbb R\times\mathbb R^n,\mathbb R)$ and $\nabla W(t,u)$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming that $L$ is bounded in the sense that there are constants $0\lt \tau_1\lt \tau_2$ such that $\tau_1 |u|^2\leq (L(t)u,u)\leq \tau_2 |u|^2$ for all $(t,u)\in \mathbb R\times \mathbb R^n$ and $W(t,u)$ is of subquadratic growth at infinity, we are able to establish two new criteria to guarantee the existence and multiplicity of classical homoclinic solutions for (HS), respectively. Recent results in the literature are extended and significantly improved.

Article information

Source
Taiwanese J. Math., Volume 18, Number 4 (2014), 1089-1105.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706478

Digital Object Identifier
doi:10.11650/tjm.18.2014.3508

Mathematical Reviews number (MathSciNet)
MR3245431

Zentralblatt MATH identifier
1357.34082

Subjects
Primary: 34C37: Homoclinic and heteroclinic solutions 35A15: Variational methods 35B38: Critical points

Keywords
homoclinic solutions critical point variational methods genus

Citation

Zhang, Ziheng; Xiang, Tian; Yuan, Rong. HOMOCLINIC SOLUTIONS FOR SUBQUADRATIC HAMILTONIAN SYSTEMS WITHOUT COERCIVE CONDITIONS. Taiwanese J. Math. 18 (2014), no. 4, 1089--1105. doi:10.11650/tjm.18.2014.3508. https://projecteuclid.org/euclid.twjm/1499706478


Export citation

References

  • C. O. Alves, P. C. Carrião and O. H. Miyagaki, Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation, Appl. Math. Lett., 16(5) (2003), 639-642.
  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14(4) (1973), 349-381.
  • P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign, Comm. Appl. Nonlinear Anal., 1(2) (1994), 97-129.
  • V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4(4) (1991), 693-727.
  • A. Daouas, Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption, Nonlinear Anal., 74(11) (2011), 3407-3418.
  • Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25(11) (1995), 1095-1113.
  • Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and supequadratic Hamiltonian systems, Nonlinear Anal., 71(5-6) (2009), 1395-1413.
  • M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations, 219(2) (2005), 375-389.
  • P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems, Electron. J. Differential Equations, (1994), 1-10.
  • X. Lv and J. Jiang, Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials, Nonlinear Analysis: Real World Applications, 13(3) (2012), 1152-1158.
  • X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 72(1) (2010), 390-398.
  • Y. Lv and C. Tang, Existence of even homoclinic orbits for a class of Hamiltonian systems, Nonlinear Anal., 67(7) (2007), 2189-2198.
  • W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5(5) (1992), 1115-1120.
  • H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Pairs, 1897-1899.
  • P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: CBMS Reg. Conf. Ser. in. Math., Vol. 65, American Mathematical Society, Provodence, RI, 1986.
  • P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A., 114(1-2) (1990), 33-38.
  • P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206(3) (1991), 473-499.
  • A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, Nonlinear Anal., 30(8) (1997), 4849-4857.
  • J. Sun, H. Chen and J. J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 373(1) (2011), 20-29.
  • X. Tang and X. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite subquadratic potentials, Nonlinear Anal., 74(17) (2011), 6314-6325.
  • X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A., 141(5) (2011), 1103-1119.
  • L. Wan and C. Tang, Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition, Discrete. Cont. Dyn. Syst. Ser. B, 15(1) (2011), 255-271.
  • J. Wang, J. Xu and F. Zhang, Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials, Comm. Pure. Appl. Anal., 10(1) (2011), 269-286.
  • J. Wang, F. Zhang and J. Xu, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 366(2) (2010), 569-581.
  • M. Yang and Z. Han, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials, Nonlinear Analysis: Real World Applications, 12(5) (2011), 2742-2751.
  • R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order Hamiltonian systems, Results. Math., 61 (2012), 195-208.
  • Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems, Nonlinear Anal., 72(2) (2010), 894-903.
  • Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems, Nonlinear Anal., 71(9) (2009), 4125-4130.
  • Z. Zhang and R. Yuan, Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems, Nonlinear Analysis: Real World Applications, 11(5) (2010), 4185-4193.
  • W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16(8) (2003), 1283-1287.