Conley and Easton established a topological theory of isolating invariant sets. It is essential to study dynamics on the boundary of isolating neighborhood. In this talk, we show the existence of orbits from the boundary to it by using the variational method. As a result, we estimate Cech cohomology of the isolating invariant sets. We apply it to potential systems with saddle-center equilibrium points to get an isolating invariant set with non-zero Cech cohomology.