The heteroclinic connection problem for general double-well potentials

By variational methods, we provide a simple proof of existence of a heteroclinic orbit to the Hamiltonian system $u^{\mathrm{\prime }\mathrm{\prime }}=\mathrm{\nabla }W(u)$ that connects the two global minima of a double-well potential $W$. Moreover, we consider several inhomogeneous extensions.