  L\'evy-type perpetuities being the a.s. limits of particular generalized
Ornstein-Uhlenbeck processes are a natural continuous-time generalization of
discrete-time perpetuities. These are random variables of the form
$S:=\int_{[0,\infty)}e^{-X_{s-}}{\mathrm{d}}Z_s$, where $(X,Z)$ is a
two-dimensional L\'evy process, and $Z$ is a drift-free L\'evy process of
bounded variation. We prove an ultimate criterion for the finiteness of power
moments of $S$. This result and the previously known assertion due to Erickson
and Maller (2005) concerning the a.s. finiteness of $S$ are then used to derive
ultimate necessary and sufficient conditions for the $L_p$-convergence for
$p>1$ and $p=1$, respectively, of Biggins' martingales associated to branching
L\'evy processes. In particular, we provide final versions of results obtained
recently by Bertoin and Mallein (2018).
