Topological Methods in Nonlinear Analysis

Almost homoclinic solutions for the second order Hamiltonian systems

Joanna Janczewska

Full-text: Open access

Abstract

The second order Hamiltonian system $\ddot{q}+V_{q}(t,q)=f(t)$, where $t\in\mathbb R$ and $q\in\mathbb R^n$, is considered. We assume that a potential $V\in C^{1}(\mathbb R\times\mathbb R^n,\mathbb R)$ is of the form $V(t,q)=-K(t,q)+W(t,q)$, where $K$ satisfies the pinching condition and $W_{q}(t,q)=o(|q|)$, as $|q|\to 0$ uniformly with respect to $t$. It is also assumed that $f\in C(\mathbb R,\mathbb R^n)$ is non-zero and sufficiently small in $L^{2}(\mathbb R,\mathbb R^n)$. In this case $q\equiv 0$ is not a solution. Therefore there are no orbits homoclinic to $0$ in a classical sense. However, we show that there is a solution emanating from $0$ and terminating at $0$. We are to call such a solution almost homoclinic to $0$. It is obtained here as a weak limit in $W^{1,2}(\mathbb R,\mathbb R^n)$ of a sequence of almost critical points.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 32, Number 1 (2008), 131-137.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463150467

Mathematical Reviews number (MathSciNet)
MR2466807

Zentralblatt MATH identifier
1223.37076

Citation

Janczewska, Joanna. Almost homoclinic solutions for the second order Hamiltonian systems. Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 131--137. https://projecteuclid.org/euclid.tmna/1463150467


Export citation

References

  • A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems , Rend. Sem. Mat. Univ. Padova, 89 (1993), 177–194 \ref\key 2
  • F. Antonacci and P. Magrone, Second order nonautonomous systems with symmetric potential changing sign , Rend. Mat. Appl. (7), 18 (1998), 367–379 \ref\key 3
  • V. Coti Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems , Math. Ann., 288 (1990), 133–160 \ref\key 4
  • V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials , J. Amer. Math. Soc., 4 (1991), 693–727 \ref\key 5
  • Y. Ding and M. Girardi, Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign , Dynam. Systems Appl., 2 (1993), 131–145 \ref\key 6
  • Y. Ding and S. J. Li, Homoclinic orbits for first order Hamiltonian systems , J. Math. Anal. Appl., 189 (1995), 585–601 \ref\key 7
  • H. Hofer and K. Wysocki, First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems , Math. Ann., 228 (1990), 483–503 \ref\key 8
  • M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems , J. Differential Equations, 219 (2005), 375–389 \ref\key 9 ––––, Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential , J. Math. Anal. Appl., 335 (2007), 1119–1127 \ref\key 10
  • J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems , Applied Mathematical Sciences, 74 , Springer–Verlag, New York (1989) \ref\key 11
  • P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems , Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33–38 \ref\key 12
  • P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems , Math. Z., 206 (1991), 473–499 \ref\key 13
  • E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems , Math. Z., 209 (1992), 27–42 \ref\key 14