## 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 $$t\in\mathbb R$$, $$u\in\mathbb R^N$$, $$V\in C^1(\mathbb R\times\mathbb R^N,\mathbb R)$$, and $$f:\mathbb R\to\mathbb R^N$$. 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 $$f_k:\mathbb R\to\mathbb R^N$$ is a $$2kT$$-periodic extension of $$f$$ to the interval $$[-kT,kT),k\in\mathbb N.$$ The main results are the following.
Theorem 1.1. Suppose that $$V$$ and $$f\neq 0$$ satisfies the following conditions
$$(H1)$$
$$V(t,x)=-K(t,x)+W(t,x)$$ is $$T-$$periodic with respect to $$t,T>0$$
$$(H2')$$
There exist constants $$b>0$$ and $$\gamma\in [1,2]$$ such that
$K(t,0)=0,K(t,x)\geq b{|x|}^\gamma \text{ for all } (t,x)\in [0,T]\times\mathbb R^N;$
$$(H3')$$
There exists a constant $$\varrho\in[2,\mu]$$ such that
$(x,\nabla K(t,x))\leq\varrho K(t,x)\text{ for all } (t,x)\in [0,T]\times\mathbb R^N;$
$$(H4)$$
$$\nabla W(t,x)=o(|x|)$$ as $$|x|\to 0$$ uniformly with respect to $$t$$;
$$(H5)$$
There is a constant $$\mu>2$$ such that for all $$(t,x)\in\mathbb R\times({\mathbb R^N}\setminus 0)$$
$0<\mu W(t,x)\leq(x,\nabla W(t,x));$
$$(H6)$$
$$f:\mathbb R\to\mathbb R^N$$ is a continuous and bounded function.
$$(H7')$$
$$\int_{\mathbb R}|f(t)|^2dt<2(\min\{\frac{\delta}{2},b\delta^{\gamma-1}-M\delta^{\mu-1}\})^2,$$ where $M=\sup\{W(t,x)|t\in[0,T], x\in\mathbb R^N,|x|=1\}$
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.
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
$K(t,x)\leq(x,\nabla K(t,x))\leq\varrho K(t,x)\text{ for }(t,x)\in [0,T]\times\mathbb R^N;$
Then system $$(HS)$$ possesses a nontrivial homoclinic solution.

### MSC:

 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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### References:

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