Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero. (English) Zbl 1218.37081
The authors consider the following first-order Hamiltonian system
\[
\dot{u}(t)=\mathcal{J}H_u(t,u), t\in \mathbb{R},
\]
where \(u=(y,z)\in \mathbb{R}^{2N}, \mathcal{J}\) is the standard symplectic matrix in \(\mathbb{R}^{2N}\), and \(H\in C^1(\mathbb{R} \times \mathbb{R}^{2N},\mathbb{R})\) has the form \(H(t,u)=\frac{1}{2}Lu \cdot u +W(t,u)\) with \(L\) being a \(2N\times 2N\) symmetric constant matrix, and \(W\in C^1(\mathbb{R} \times \mathbb{R}^{2N},\mathbb{R})\). The main result of the paper shows, by using two recent critical point theorems for strongly indefinite functionals [T. Bartsch and Y. Ding, Math. Nachr. 279, No. 12, 1267–1288 (2006; Zbl 1117.58007)], that if the technical working assumptions \((L_1)\), \((H_1) -(H_5)\) hold, then the considered Hamiltonian system has at least one homoclinic orbit (Theorem 1.1).
Reviewer: Dorin Andrica (Riyadh)
MSC:
| 37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
Keywords:
homoclinic orbits; Hamiltonian systems; asymptotically linear terms; variational methods; strongly indefinite problemsCitations:
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