  The aim of this article is to study a Cahn-Hilliard model for a
multicomponent mixture with cross-diffusion effects, degenerate mobility and
where only one of the species does separate from the others. We define a notion
of weak solution adapted to possible degeneracies and our main result is
(global in time) existence. In order to overcome the lack of a-priori
estimates, our proof uses the formal gradient flow structure of the system and
an extension of the boundedness by entropy method which involves a careful
analysis of an auxiliary variational problem. This allows to obtain solutions
to an approximate, time-discrete system. Letting the time step size go to zero,
we recover the desired weak solution where, due to their low regularity, the
Cahn-Hilliard terms require a special treatment.
