Heteroclinic connections between nonconsecutive equilibria of a fourth order differential equation

Assuming that f is a potential having three minima at the same level of energy, we study for the conservative equation uiv - g(u)u″ - 1/2g′(u)u′2 + f′(u) = 0, (1) the existence of a heteroclinic connection between the extremal equilibria. Our method consists in minimizing the functional ∫-∞ +∞[1/2[(u″2) + g(u)u′2] + f(u)] dx whose Euler-Lagrange equation is given by (1), in a suitable space of functions.