Multiple homoclinics for a class of singular Hamiltonian systems

We consider a Hamiltonian system $\ddot u+\nabla V(u)=0$ where the potential $V\colon\mathbb{R}^N\setminus S\to\mathbb{R}$ has a unique strict global maximum at a point $p\in\mathbb{R}^N$ and a singular set $S\subset\mathbb{R}^N\setminus\{p\}$ such that $\mathbb{R}^N\setminus S$ is open, path-connected and the fundamental group $G=\pi_1(\mathbb{R}^N\setminus S)$ is non trivial. Under some compactness conditions on $V$ at infinity and around the singular set $S$ we study the existence of homoclinic orbits to $p$ which link with $S$. When $V$ and $G$ satisfy suitable geometrical conditions, we can prove the existence of multiple homoclinics, each one belonging to a different homotopy class of $G$.