  Bound states in the continuum (BIC) are of special interest in photonics due
to their theoretically infinite radiative lifetime. Here, we design a structure
composed of a GaN layer with guided exciton-polaritons and a TiO$_2$ 1D
photonic crystal slab. The photonic BIC hosted by the photonic crystal slab
couples with the excitons of GaN to form a polaritonic BIC with a negative
mass. This allows condensation to be reached with a low threshold in a
structure suitable for electrical injection, paving the way for
room-temperature polariton microdevices. We study in detail how the repulsive
interaction between exciton-polaritons affects the condensate distribution in
reciprocal space and, consequently, the condensate's overlap with the BIC
resonance and, therefore, the condensate lifetime. We study an intrinsic
contribution related to the formation of a bright soliton and the extrinsic
contribution related to the interaction with an excitonic reservoir induced by
spatially focused non-resonant pumping. We then study the peculiar dynamics of
the condensation process in a BIC state for interacting particles using
Boltzmann equations and hybrid Boltzmann-Gross Pitaevskii equations. We find
optimal conditions allowing one to benefit from the long lifetime of the BIC
for polariton condensation in a real structure.
