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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 $z=(p,q)\in \Bbb R^{2N}$ and $J=\left (\smallmatrix 0 & -I\\ I & 0\endsmallmatrix \right)$. 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 $x\in \Bbb R^N$, $V\in C(\Bbb R^n,\Bbb R)$ and $f\in C(\Bbb R^N,\Bbb R)$.

37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
Full Text: DOI
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