Abstract
Abstract Given a C2 nonnegative potential F with a discrete set of zeros we consider the existence of heteroclinics connecting two arbitrary zeros of F for the Swift-Hohenberg equation, i.e. uʺʺ− βuʺ + Fʹ (u) = 0, for β ≤ 0. The emphasis is on the case where F has more than two zeros. We prove that when Fʹ has a sign near the zeros of F, |β| is sufficiently small in a variational sense, and then given any two different zeros z– and z+, there exists a solution to the Swift-Hohenberg equation with boundary values . We also consider heteroclinics constrained in a strip of the form {l < u(x) < r}, and consider some local conditions on F which produce heteroclinics regardless of the behavior of F outside [l, r].
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