Backward stochastic differential equations with random stopping time and singular final condition

In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: $$Y_t=\xi -{\int }_{t\wedge \tau }^{\tau }{Y}_r\mathrm{\mid }{Y}_r{\mathrm{\mid }}^{q}dr-{\int }_{t\wedge \tau }^{\tau }{Z}_{r}d{B}_r,t\ge 0,$$ where $\tau$ is a stopping time, $q$ is a positive constant and $\xi$ is a ${\mathcal{F}}_{\tau }$-measurable random variable such that $\mathbf{P}(\xi =+\mathrm{\infty })>0$. We study the link between these BSDE and the Dirichlet problem on a domain $D\subset {\mathbb{R}}^d$ and with boundary condition $g$, with $g=+\mathrm{\infty }$ on a set of positive Lebesgue measure. We also extend our results for more general BSDE.