Heteroclinic solutions for the extended Fisher–Kolmogorov equation

We study the extended Fisher–Kolmogorov (EFK) equation and its variants. By variational approach, we show that, for every global minimum ξ of the potential function V , there is a pair of heteroclinic solutions, one emanating from ξ and the other terminating at ξ . We require neither superquadratic growth of V nor the presence of saddle-focus equilibria; moreover V is allowed to approach its minimal level near infinity.