  Three non-existence results are established for self-gravitating solitons in
Einstein-Maxwell-scalar models, wherein the scalar field is, generically,
non-minimally coupled to the Maxwell field via a scalar function $f(\Phi)$.
Firstly, a trivial Maxwell field is considered, which yields a consistent
truncation of the full model. In this case, using a scaling (Derrick-type)
argument, it is established that no stationary and axisymmetric
self-gravitating scalar solitons exist, unless the scalar potential energy is
somewhere negative in spacetime. This generalises previous results for the
static and strictly stationary cases. Thus, rotation alone cannot support
self-gravitating scalar solitons in this class of models. Secondly, constant
sign couplings are considered. Generalising a previous argument by Heusler for
electro-vacuum, it is established that no static self-gravitating
electromagnetic-scalar solitons exist. Thus, a varying (but constant sign)
electric permittivity alone cannot support static Einstein-Maxwell-scalar
solitons. Finally, the second result is generalised for strictly stationary,
but not necessarily static, spacetimes, using a Lichnerowicz-type argument,
generalising previous results in models where the scalar and Maxwell fields are
not directly coupled. The scope of validity of each of these results points out
the possible paths to circumvent them, in order to obtain self-gravitating
solitons in Einstein-Maxwell-scalar models.
