  The central object of this article is (a special version of) the Helfrich
functional which is the sum of the Willmore functional and the area functional
times a weight factor $\varepsilon\ge 0$. We collect several results concerning
the existence of solutions to a Dirichlet boundary value problem for Helfrich
surfaces of revolution and cover some specific regimes of boundary conditions
and weight factors $\varepsilon\ge 0$. These results are obtained with the help
of different techniques like an energy method, gluing techniques and the use of
the implicit function theorem close to Helfrich cylinders. In particular,
concerning the regime of boundary values, where a catenoid exists as a global
minimiser of the area functional, existence of minimisers of the Helfrich
functional is established for \emph{all} weight factors $\varepsilon\ge 0$. For
the singular limit of weight factors $ \varepsilon \nearrow \infty $ they
converge uniformly to the catenoid which minimises the surface area in the
class of surfaces of revolution.
