Asymptotic behavior of positive solutions of a semilinear Dirichlet problem

Abstract

In this paper, we study the asymptotic behavior of the unique positive classical solution to the following semilinear boundary value problem Δ u + a ( x ) u α = 0 , x ∈ Ω , u > 0 in Ω , u | ∂ Ω = 0 . Here Ω is a bounded C 1 , 1 domain, α < 1 and the function a is in C l o c γ ( Ω ) , 0 < γ < 1 such that there exists c > 0 satisfying for each x ∈ Ω , 1 c ≤ a ( x ) δ ( x ) λ exp ( − ∫ δ ( x ) η z ( t ) t d t ) ≤ c , where λ ≤ 2 , η > d = d i a m ( Ω ) , δ ( x ) = d i s t ( x , ∂ Ω ) and z is a continuous function on [ 0 , η ] with z ( 0 ) = 0 .