The exact asymptotic behaviour of the unique solution to a singular nonlinear Dirichlet problem

Abstract

By Karamata regular varying theory, a perturbed argument and constructing comparison functions, we show the exact asymptotic behaviour of the unique solution u ∈ C 2 ( Ω ) ∩ C ( Ω ¯ ) near the boundary to a singular Dirichlet problem − Δ u = b ( x ) g ( u ) + λ f ( u ) , u > 0 , x ∈ Ω , u | ∂ Ω = 0 , which is independent on λ f ( u ) , and we also show the existence and uniqueness of solutions to the problem, where Ω is a bounded domain with smooth boundary in R N , λ > 0 , g ∈ C 1 ( ( 0 , ∞ ) , ( 0 , ∞ ) ) and there exists γ > 1 such that lim t → 0 + g ′ ( ξ t ) g ′ ( t ) = ξ − ( 1 + γ ) , ∀ ξ > 0 , f ∈ C loc α ( [ 0 , ∞ ) , [ 0 , ∞ ) ) , the function f ( s ) s + s 0 is decreasing on ( 0 , ∞ ) for some s 0 > 0 , and b is nonnegative nontrivial on Ω, which may be vanishing on the boundary.