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Homoclinic solutions for a class of second-order Hamiltonian systems. (English) Zbl 1185.34056
This paper considers the second-order Hamiltonian systems $$\ddot{u}(t)+\nabla V(t,u(t))=f(t),\tag HS$$ where $t\in\Bbb R$, $u\in\Bbb R^N$, $V\in C^1(\Bbb R\times\Bbb R^N,\Bbb R)$, and $f:\Bbb R\to\Bbb R^N$. The authors prove the existence of a homoclinic solution of $(HS)$ as the limit of $2KT$-periodic solutions of $$\ddot{u}(t)=-\nabla V(t,u(t))+f_k(t),\tag HS_k$$ where $f_k:\Bbb R\to\Bbb R^N$ is a $2kT$-periodic extension of $f$ to the interval $[-kT,kT),k\in\Bbb N.$ The main results are the following. Theorem 1.1. Suppose that $V$ and $f\neq 0$ satisfies the following conditions {\parindent=11mm \item{$(H1)$} $V(t,x)=-K(t,x)+W(t,x)$ is $T-$periodic with respect to $t,T>0$ \item{$(H2')$} There exist constants $b>0$ and $\gamma\in [1,2]$ such that $$K(t,0)=0,K(t,x)\geq b{|x|}^\gamma \text{ for all } (t,x)\in [0,T]\times\Bbb R^N;$$ \item{$(H3')$} There exists a constant $\varrho\in[2,\mu]$ such that $$(x,\nabla K(t,x))\leq\varrho K(t,x)\text{ for all } (t,x)\in [0,T]\times\Bbb R^N;$$ \item{$(H4)$} $\nabla W(t,x)=o(|x|)$ as $|x|\to 0$ uniformly with respect to $t$; \item{$(H5)$} There is a constant $\mu>2$ such that for all $(t,x)\in\Bbb R\times({\Bbb R^N}\setminus 0)$ $$0<\mu W(t,x)\leq(x,\nabla W(t,x));$$ \item{$(H6)$} $f:\Bbb R\to\Bbb R^N$ is a continuous and bounded function. \item{$(H7')$} $\int_{\Bbb R}|f(t)|^2dt<2(\min\{\frac{\delta}{2},b\delta^{\gamma-1}-M\delta^{\mu-1}\})^2,$ where $$M=\sup\{W(t,x)|t\in[0,T], x\in\Bbb R^N,|x|=1\}$$ and $\delta \in (0,1]$ such that $$b\delta^{\gamma-1}-M\delta^{\mu-1}=\max_{x\in[0,1]}(bx^{\gamma-1}-Mx^{\mu-1}).$$ Then system $(HS)$ possesses a nontrivial homoclinic solution. \par} Theorem 1.2. Suppose that $V$ and $f=0$ satisfies $(H1)$, $(H2')$, $(H4)-(H6)$ and the following $(H3'')$ There exists a constant $\varrho\in[2,\mu]$ such that $$K(t,x)\leq(x,\nabla K(t,x))\leq\varrho K(t,x)\text{ for }(t,x)\in [0,T]\times\Bbb R^N;$$ Then system $(HS)$ possesses a nontrivial homoclinic solution.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions of ODE 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
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##### References:
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