## Homoclinic solutions for a second-order nonperiodic asymptotically linear Hamiltonian systems.(English)Zbl 1268.37080

The second-order Hamiltonian system $\ddot{u}(t)+\nabla V(t,u(t))=0$ is studied, where $$V(t,u)=-K(t,u)+W(t,u)$$ and $$K,W:\mathbb R \times \mathbb R ^N\to \mathbb R$$ are $$C^1$$-maps. The main result is the following theorem:
Let $$A:=\sup \{ K(t,u): t\in \mathbb R, |u|\leq 1\} <+\infty$$. Moreover, assume the following:
1) $$K(t,0)\equiv 0$$, and there is a constant $$a>0$$ such that $K(t,u)\geq a|u|^2, \quad \forall (t,u)\in \mathbb R \times \mathbb R^N;$
2) there exists $$\beta \in (1,2]$$ such that $K(t,u)\leq (u,\nabla K(t,u))\leq \beta K(t,u),\quad \forall (t,u)\in \mathbb R \times \mathbb R^N;$
3) $$W(t,0)\equiv 0$$, and $$\nabla W(t,u)=o(|u|)$$ as $$u\to 0$$ uniformly in $$t$$, and there exists $$M_0>0$$ such that $\frac{|\nabla W(t,u)|}{|u|}\leq M_0, \quad\forall (t,u)\in \mathbb R \times \mathbb R^N;$ 4) $$W(t,u)-w(t)|u|^2=o(|u|^2)$$ as $$|u|\to \infty$$ uniformly in $$t$$, where $$w\in L^\infty(\mathbb R, \mathbb R)$$ with $$w_\infty :=\inf_{t\in \mathbb R}w(t)>2A;$$
5) $$\widetilde{W}(t,u):=1/2(\nabla W(t,u),u)-W(t,u)\to +\infty$$ as $$|u|\to +\infty$$, and $\inf \left\{ \frac{\widetilde{W}(t,u)}{|u|^2}: t\in \mathbb R , \quad c\leq |u|<d \right\} ,$ for any $$c,d>0$$.
Then the above system possesses a nontrivial homoclinic solution $$u\in W^{1,2}(\mathbb R, \mathbb R^N)$$ such that $$\dot{u}(t)\to 0$$ as $$t\to \pm \infty$$.
The author was motivated by a recent result of A. Daouas [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 11, 3407–3418 (2011; Zbl 1221.37115)] dealing with superquadratic $$W$$. Results of M. Izydorek and J. Janczewska [J. Differ. Equations 219, No. 2, 375–389 (2005; Zbl 1080.37067)] are used in the proof.

### MSC:

 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37C29 Homoclinic and heteroclinic orbits for dynamical systems

### Keywords:

second order Hamiltonian systems; homoclinic solutions

### Citations:

Zbl 1221.37115; Zbl 1080.37067
Full Text:

### References:

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