In this paper, we give an exact asymptotic of the unique solution to the following singular boundary value problem −Δu=a(x)g(u),x∈Ω,u>0, in Ω,u|∂Ω=0. Here Ω is a C2-bounded domain in Rn(n≥2), g∈C1((0,∞),(0,∞)) is nonincreasing on (0,∞) with limt→0g′(t)∫0tdsg(s)=−Cg≤0 and the function a is in Clocα(Ω), 0<α<1 satisfying 0<a1=lim infd(x)→0a(x)h(d(x))≤lim supd(x)→0a(x)h(d(x))=a2<∞, where h(t)=ct−λexp(∫tηz(s)sds), λ≤2, c>0 and z continuous on [0,η] for some η>0 such that z(0)=0. Two applications of this result are also given. The first concerns the boundary behavior of the unique solution of −Δu+βu|∇u|2=a(x)g(u) that vanishes on the boundary and the second concerns the behavior of u in the case where the open set Ω is an annular and the behaviors of the function a on the interior boundary and the exterior boundary may be different.