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

We consider an open connected set Ω and a smooth potential U which is positive in Ω and vanishes on ∂Ω. We study the existence of orbits of the mechanical system u¨=Ux(u), that connect different components of ∂Ω and lie on the zero level of the energy. We allow that ∂Ω contains a finite number of critical points of U. The case of symmetric potential is also considered.