Existence and multiplicity of homoclinic orbits for potentials on unbounded domains

We study the system $\ddot q=-V'(q)$ in $\mathbb{R}^N$ where $V$ is a potential with a strict local maximum at 0 and possibly with a singularity. First, using a minimizing argument, we can prove existence of a homoclinic orbit when the component $\Omega$ of $\{x\in\mathbb{R}^N:\,V(x)<V(0)\}$ containing 0 is an arbitrary open set; in the case $\Omega$ unbounded we allow $V(x)$ to go to 0 at infinity, although at a slow enough rate. Then, we show that the presence of a singularity in $\Omega$ implies that a homoclinic solution can be found also via a min--max procedure and, comparing the critical levels of the functional associated to the system, we see that the two solutions are distinct whenever the singularity is ``not too far'' from 0.