Exact boundary behavior of the unique positive solution to some singular elliptic problems

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 00 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.