Asymptotic behavior of positive solutions of a nonlinear Dirichlet problem

Abstract

We take up the existence and the asymptotic behavior of a classical solution to the following semilinear Dirichlet problem {−Δu=a(x)g(u),x∈Ω,u>0in Ω,u|∂Ω=0, where Ω is a C1,1-bounded domain in RN, N≥2 and the function a belongs to Clocγ(Ω), (0<γ<1) such that there exist c1,c2>0 satisfying for each x∈Ω, c1δ(x)−λ1exp(∫δ(x)ηz1(s)sds)≤a(x)≤c2δ(x)−λ2exp(∫δ(x)ηz2(s)sds), where η>diam(Ω), δ(x)=dist(x,∂Ω), λ1≤λ2≤2 and for i∈{1,2}, zi is a continuous function on [0,η] with zi(0)=0.Our arguments are based on the sub–supersolution method with Karamata regular variation theory.