  We investigate and characterize the emergence of finite-component dissipative
phase transitions (DPTs) in nonlinear photon resonators subject to $n$-photon
driving and dissipation. Exploiting a semiclassical approach, we derive general
results on the occurrence of second-order DPTs in this class of systems. We
show that for all odd $n$, no second-order DPT can occur while, for even $n$,
the competition between higher-order nonlinearities determines the nature of
the criticality and allows for second-order DPTs to emerge only for $n=2$ and
$n=4$. As pivotal examples, we study the full quantum dynamics of three- and
four-photon driven-dissipative Kerr resonators, confirming the prediction of
the semiclassical analysis on the nature of the transitions. The stability of
the vacuum and the typical timescales needed to access the different phases are
also discussed. We also show a first-order DPT where multiple solutions emerge
around zero, low, and high-photon numbers. Our results highlight the crucial
role played by strong and weak symmetries in triggering critical behaviors,
providing a Liouvillian framework to study the effects of high-order nonlinear
processes in driven-dissipative systems, that can be applied to problems in
quantum sensing and information processing.
