Exact boundary behavior of positive large solutions of a nonlinear Dirichlet problem. (October 2019) - Khamessi, Bilel; Ben Othman, Sonia
Abstract: In this paper, we investigate the exact asymptotic behavior of positive solution to the following singular boundary value problem Δ u = b ( x ) f ( u ), x ∈ Ω, u > 0  in  Ω, u | ∂ Ω = + ∞, where Ω is a C 2 -bounded domain in R N, ( N ≥ 3 ), f ∈ C 1 ( ( 0, ∞ ), ( 0, ∞ ) ) is nondecreasing on ( 0, ∞ ) and b is a function in C l o c γ ( Ω ), ( 0 < γ < 1 ) such that there exist b 1, b 2 > 0 satisfying for each x ∈ Ω, 0 < b 1 = lim d ( x ) ⟶ 0 inf b ( x ) h ( d ( x ) ) ≤ lim d ( x ) ⟶ 0 sup b ( x ) h ( d ( x ) ) = b 2 < ∞, where d ( x ) = d i s t ( x, ∂ Ω ) and h ( t ) ≔ c t − λ exp ∫ t η z ( s ) s d s, η > d i a m ( Ω ), λ ≤ 2 such that z is a continuous function on [ 0, η ] with z ( 0 ) = 0 .