The authors consider the following dynamical system \[ \begin{aligned} \frac {d}{dt}\bigl (| \dot {u}_1(t) | ^{q-2} \dot {u}_1(t) \bigr)&= \nabla_{u_1} F\bigl (t,u_1(t),u_2(t)\bigr) \qquad \text{for a.e.}\quad t\in [0,T],\\ \frac {d}{dt}\bigl (| \dot {u}_2(t) | ^{p-2} \dot {u}_2(t) \bigr)&= \nabla_{u_2} F\bigl (t,u_1(t),u_2(t)\bigr) \qquad \text{for a.e.}\quad t\in [0,T],\\ u_1(0)-u_1(T) &= \dot {u}_1(0)-\dot {u}_1(T) = 0,\\ u_2(0)-u_2(T) &= \dot {u}_2(0)-\dot {u}_2(T)=0, \end{aligned} \] where, \(1<p\), \(q <\infty \), \(T>0\) and \(| \cdot | \) denotes the Euclidean norm in \(\mathbb {R}^N\). \(F\:[0,T]\times \mathbb {R}^N \times \mathbb {R}^N\to \mathbb {R}\) satisfies the following assumption:
(A) \(F\) is measurable in \(t\) for each \((x_1, x_2)\in \mathbb {R}^N\times \mathbb {R}^N\) and continuously differentiable in \((x_1, x_2)\) for a.e. \(t\in [0,T]\), and there exist \(a_1, a_2 \in C( \mathbb {R}_+, \mathbb {R}_+)\) and \(b \in L^1(0,T; \mathbb {R}_+)\) such that \[ \Bigl | F(t,x_1,x_2)\Bigr |, \Bigr | \nabla_{x_1}F(t,x_1,x_2)\Bigr |, \Bigr | \nabla_{x_2}F(t,x_1,x_2)\Bigr | \leq \Bigr [a_1\bigr (| x_1| \bigl) + a_2\bigr (| x_2| \bigl)\Bigl]b(t) \] for all \((x_1, x_2)\in \mathbb {R}^N\times \mathbb {R}^N\) and a.e. \(t\in [0,T]\).
By using the least action principle and the saddle point theorem, two existence results are proved in this paper, which generalize some known results of D. Paşca and C. L. Tang [Appl. Math. Lett. 23, No. 3, 246–251, (2010; Zbl 1187.34051)].