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Zbl 1218.37081
Sun, Juntao; Chen, Haibo; Nieto, Juan J.
Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero.
(English)
[J] J. Math. Anal. Appl. 378, No. 1, 117-127 (2011). ISSN 0022-247X

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 [{\it T. Bartsch} and {\it 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).
MSC 2000:
*37J45 Periodic, homoclinic and heteroclinic orbits, etc.
34C37 Homoclinic and heteroclinic solutions of ODE
58E05 Abstract critical point theory

Keywords: homoclinic orbits; Hamiltonian systems; asymptotically linear terms; variational methods; strongly indefinite problems

Citations: Zbl 1117.58007

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