Existence and global behavior of positive solution for semilinear problems with boundary blow-up

Using the sub-supersolution method with Karamata regular variation theory, we study the existence and asymptotic behavior of a classical solution to the following boundary blow-up semilinear Dirichlet problem{Δu=a(x)f(u),x∈Ω,u>0in Ω;limδ(x)→0⁡u(x)=∞, where Ω   is a C1,1C1,1-bounded domain in RnRn, n≥2n≥2 and the function a   belongs to Clocα(Ω), (0<α<1)(0<α<1) such that for each x∈Ωx∈Ω,c1(δ(x))−λ1exp⁡(∫δ(x)ηz1(s)sds)≤a(x)≤c2(δ(x))−λ2exp⁡(∫δ(x)ηz2(s)sds), where η>diam(Ω)η>diam(Ω), c1>0c1>0, c2>0c2>0, δ(x)=dist(x,∂Ω)δ(x)=dist(x,∂Ω), λ1≤λ2≤2λ1≤λ2≤2 and for i∈{1,2}i∈{1,2}, zizi is a continuous function on [0,η][0,η] with zi(0)=0zi(0)=0. For the function f  , we assume that there exist constants k1k1, k2k2, p1p1, p2p2 with 00andf(t)≥k1tp1for t≥1.