Heteroclinic connections between nonconsecutive equilibria of a fourth order differential equation. (English) Zbl 1031.34043

Bonheure, D.; Sanchez, L.; Tarallo, M.; Terracini, S.

Calc. Var. Partial Differ. Equ. 17, No. 4, 341-356 (2003).

Summary: Assuming that \(f\) is a potential having three minima at the same level of energy, we study for the conservative equation \[ u^{iv}-g(u)u'' \tfrac 12 g'(u){u'}^2 + f'(u) = 0\tag{1} \] the existence of a heteroclinic connection between the extremal equilibria. Our method consists in minimizing the functional \[ \int_{-\infty}^{+\infty} \left[\tfrac 12 [({u''}^2) + g(u){u'}^2] + f(u) \right] dx \] whose Euler-Lagrange equation is given by (1), in a suitable space of functions.

MSC:

34C37	Homoclinic and heteroclinic solutions to ordinary differential equations
37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)

Keywords:

conservative equation; heteroclinic connection; extremal equilibria