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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 \(t\in\mathbb R\), \(u\in\mathbb R^N\), \(V\in C^1(\mathbb R\times\mathbb R^N,\mathbb R)\), 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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