Note on homoclinic solutions to nonautonomous Hamiltonian systems with sign-changing nonlinear part

Abstract

In the paper, we utilize the recent variational, abstract theorem to show the existence of homoclinic solutions to the Hamiltonian system $$\dot{z}=J{D}_{z}H(z,t),t\in \mathbb{R},$$ where the Hamiltonian $H:{\mathbb{R}}^{2N}\times \mathbb{R}\to \mathbb{R}$ is of the form $$H(z,t)=\frac{1}{2}Az\cdot z+\Gamma (t)\left(F(z)-\lambda G(z)\right)$$ for some symmetric matrix $A$.