Exact boundary behavior of positive large solutions of a nonlinear Dirichlet problem

Abstract

In this paper, we investigate the exact asymptotic behavior of positive solution to the following singular boundary value problem Δu=b(x)f(u),x∈Ω,u>0 in Ω,u|∂Ω=+∞,where Ω is a C2-bounded domain in RN, (N≥3), f∈C1((0,∞),(0,∞)) is nondecreasing on (0,∞) and b is a function in Clocγ(Ω), (0<γ<1) such that there exist b1,b2>0 satisfying for each x∈Ω, 0<b1=limd(x)⟶0infb(x)h(d(x))≤limd(x)⟶0supb(x)h(d(x))=b2<∞,where d(x)=dist(x,∂Ω) and h(t)≔ct−λexp∫tηz(s)sds, η>diam(Ω), λ≤2 such that z is a continuous function on [0,η] with z(0)=0.