In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: Yt=ξ−∫t∧ττYr|Yr|q dr−∫t∧ττZr dBr, t≥0, where τ is a stopping time, q is a positive constant and ξ is a ℱτ-measurable random variable such that P(ξ=+∞)>0. We study the link between these BSDE and the Dirichlet problem on a domain D⊂ℝd and with boundary condition g, with g=+∞ on a set of positive Lebesgue measure. We also extend our results for more general BSDE.