On the existence of connecting orbits for critical values of the energy

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