Asymptotic behavior of positive solutions of a nonlinear Dirichlet problem

We take up the existence and the asymptotic behavior of a classical solution to the following semilinear Dirichlet problem { − Δ u = a ( x ) g ( u ) , x ∈ Ω , u > 0 in  Ω , u | ∂ Ω = 0 , where Ω is a C 1 , 1 -bounded domain in R N , N ≥ 2 and the function a belongs to C l o c γ ( Ω ) , ( 0 < γ < 1 ) such that there exist c 1 , c 2 > 0 satisfying for each x ∈ Ω , c 1 δ ( x ) − λ 1 exp ( ∫ δ ( x ) η z 1 ( s ) s d s ) ≤ a ( x ) ≤ c 2 δ ( x ) − λ 2 exp ( ∫ δ ( x ) η z 2 ( s ) s d s ) , where η > d i a m ( Ω ) , δ ( x ) = d i s t ( x , ∂ Ω ) , λ 1 ≤ λ 2 ≤ 2 and for i ∈ { 1 , 2 } , z i is a continuous function on [ 0 , η ] with z i ( 0 ) = 0 . guments are based on the sub–supersolution method with Karamata regular variation theory.