Asymptotic behavior of positive solutions of a semilinear Dirichlet problem

In this paper, we study the asymptotic behavior of the unique positive classical solution to the following semilinear boundary value problem Δu+a(x)uα=0,  x∈Ω,  u>0 in Ω, u|∂Ω=0. Here Ω is a bounded C1,1C1,1 domain, α<1α<1 and the function aa is in Clocγ(Ω), 0<γ<10<γ<1 such that there exists c>0c>0 satisfying for each x∈Ω, 1c≤a(x)δ(x)λexp(−∫δ(x)ηz(t)tdt)≤c, where λ≤2λ≤2, η>d=diam(Ω), δ(x)δ(x)=dist(x,∂Ω) and zz is a continuous function on [0,η][0,η] with z(0)=0z(0)=0.