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 ξξ of the potential function VV, there is a pair of heteroclinic solutions, one emanating from ξξ and the other terminating at ξξ. We require neither superquadratic growth of VV nor the presence of saddle-focus equilibria; moreover VV is allowed to approach its minimal level near infinity.