## Homoclinic orbits for asymptotically linear Hamiltonian systems.(English)Zbl 0984.37072

The existence of a homoclinic orbit is proved in the paper for a Hamiltonian system $\dot z=JH_z(z,t),\tag{1}$ where $$z=(p,q)\in \mathbb R^{2N}$$ and $$J=\left (\begin{smallmatrix} 0 & -I\\ I & 0\end{smallmatrix} \right)$$. Furthermore, $$H(z,t)=\frac{1}{2}Az\cdot z+G(z,t)$$ and $$H(0,t)=0$$ with $$G_z(z,t)/|z|\to 0$$ uniformly in $$t$$ as $$z\to 0$$, $$G$$ is 1-periodic in $$t$$ and asymptotically linear at infinity, and $$JA$$ is a hyperbolic matrix. $$G$$ has additional properties. A variational method is used to get an abstract theorem which is applied for showing a homoclinic orbit of (1). That theorem is also used to show a decaying solution of an asymptotically linear Schrödinger equation $$-\triangle u+V(x)u=f(x,u)$$ for $$x\in \mathbb R^N$$, $$V\in C(\mathbb R^n,\mathbb R)$$ and $$f\in C(\mathbb R^N,\mathbb R)$$.

### MSC:

 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)