A new proof of the existence of homoclinic orbits for a class of autonomous second order Hamiltonian systems in Rn

We consider the Hamiltonian system in $\mathbb{R}^N$ given by $\ddot u+V'(u)=0$ where $V\colon\mathbb{R}^N\to\mathbb{R}$ is a smooth potential having a non degenerate local maximum at $0$ and we assume that there is an open bounded neighborhood $\Omega$ of $0$ such that $V(x)<V(0)$ for $x\in\Omega\setminus\{0\}$, $V(x)=V(0)$ and $V'(x)\ne 0$ for $x\in\partial\Omega$. Using a refined version of the mountain pass lemma due to Ghoussoub and Preiss, we give a further proof of the existence of a solution to $\ddot u+V'(u)=0$, homoclinic to 0.