Exact behavior of the unique positive solution to some singular elliptic problem in exterior domains

In this paper, we give an exact behavior on the boundary and at infinity 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  and  lim|x|→∞u(x)=0. Here ΩΩ is an exterior domain in Rn(n≥3) with compact C2C2-boundary, g∈C1((0,∞),(0,∞))g∈C1((0,∞),(0,∞)) is nonincreasing on (0,∞)(0,∞) with limt→0g′(t)∫0tdsg(s)=−Cg≤0 and the function aa is in Clocα(Ω), 0<α<10<α<1 satisfying 00c1>0 and zz is continuous on [0,η][0,η] for some η>0η>0 such that z(0)=0z(0)=0 and k(t)=c2t−μexp(∫1ty(s)sds), μ≥2μ≥2, c2>0c2>0 and yy is continuous on [1,∞)[1,∞) such that limt→∞y(t)=0limt→∞y(t)=0.