On the existence of connecting orbits for critical values of the energy (1701.07514v1)

Abstract: We consider an open connected set $\Omega$ and a smooth potential $U$ which is positive in $\Omega$ and vanishes on $\partial\Omega$. We study the existence of orbits of the mechanical system [ \ddot{u}=U_x(u), ] that connect different components of $\partial\Omega$ and lie on the zero level of the energy. We allow that $\partial\Omega$ contains a finite number of critical points of $U$. The case of symmetric potential is also considered.