  We consider a class of Backward Stochastic Differential Equations with
superlinear driver process $f$ adapted to a filtration supporting at least a
$d$ dimensional Brownian motion and a Poisson random measure on ${\mathbb R}^m-
\{0\}.$ We consider the following class of terminal conditions $\xi_1 = \infty
\cdot 1_{\{\tau_1 \le T\}}$ where $\tau_1$ is any stopping time with a bounded
density in a neighborhood of $T$ and $\xi_2 = \infty \cdot 1_{A_T}$ where
$A_t$, $t \in [0,T]$ is a decreasing sequence of events adapted to the
filtration ${\mathcal F}_t$ that is continuous in probability at $T$. A special
case for $\xi_2$ is $A_T = \{\tau_2 > T\}$ where $\tau_2$ is any stopping time
such that $P(\tau_2 =T) =0.$ In this setting we prove that the minimal
supersolutions of the BSDE are in fact solutions, i.e., they attain almost
surely their terminal values. We further show that the first exit time from a
time varying domain of a $d$-dimensional diffusion process driven by the
Brownian motion with strongly elliptic covariance matrix does have a continuous
density; therefore such exit times can be used as $\tau_1$ and $\tau_2$ to
define the terminal conditions $\xi_1$ and $\xi_2.$ The proof of existence of
the density is based on the classical Green's functions for the associated PDE.
