  We study a variational model for transition layers in thin ferromagnetic
films with an underlying functional that combines an Allen-Cahn type structure
with an additional nonlocal interaction term. The model represents the
magnetisation by a map from $\mathbb{R}$ to $\mathbb{S}^1$. Thus it has a
topological invariant in the form of a winding number, and we study minimisers
subject to a prescribed winding number. As shown in our previous paper
Ignat-Moser (JDE 2017), the nonlocal term gives rise to solutions that would
not be present for a functional including only the (local) Allen-Cahn terms. We
complete the picture here by proving existence of minimisers in all cases where
it has been conjectured. In addition, we prove non-existence in some other
cases.
