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

Abstract

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 Ω is an exterior domain in Rn(n≥3) with compact C2-boundary, 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<∞, and 0<b1=lim inf|x|→∞a(x)k(|x|)≤lim sup|x|→∞a(x)k(|x|)=b2<∞, where d(x) is the Euclidean distance from x∈Ω to the boundary ∂Ω, h(t)=c1t−λexp(∫tηz(s)sds), λ≤2, c1>0 and z is continuous on [0,η] for some η>0 such that z(0)=0 and k(t)=c2t−μexp(∫1ty(s)sds), μ≥2, c2>0 and y is continuous on [1,∞) such that limt→∞y(t)=0.