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Zbl 1216.34033
Wan, Li-Li; Tang, Chun-Lei
Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without $(AR)$ condition.
(English)
[J] Discrete Contin. Dyn. Syst., Ser. B 15, No. 1, 255-271 (2011). ISSN 1531-3492; ISSN 1553-524X/e

Consider the second order Hamiltonian system $$\ddot{u}(t)-L(t)u(t)+\nabla W(t, u(t))=0,\quad t\in \mathbb{R},$$ where $L\in C(\mathbb{R}, \mathbb{R}^{N^2})$ is a symmetric matrix valued function, $W\in C^1(\mathbb{R}\times \mathbb{R}^N, \mathbb{R})$ and $\nabla W(t, x)$ denotes the gradient of $W$ with respect to $x$. As usual, a solution $u$ of this problem is homoclinic to $0$ if $u\in C^2(\mathbb{R}, \mathbb{R}^N)$ and $u(t)\rightarrow 0$ as $|t|\rightarrow 0$. The well-known Ambrosetti-Rabinowitz (AR) condition states that there exists a constant $\mu >2$ such that $0<\mu W(t, x)\leq (\nabla W(t, x), x)$ for all $t\in \mathbb{R}$ and all $x\in \mathbb{R}^N\setminus \{0\}$. In this paper, the existence of homoclinic orbits is obtained without the (AR) condition but using the concentration-compactness principle and the fountain theorem.
[Mircea Crâşmăreanu (Iaşi)]
MSC 2000:
*34C37 Homoclinic and heteroclinic solutions of ODE
37J45 Periodic, homoclinic and heteroclinic orbits, etc.
47J30 Variational methods

Keywords: homoclinic orbits; second order Hamiltonian systems; concentration compactness principle; fountain theorem

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