ABSTRACT
Let the positive constants ε, m, k, s, A, B 0, R, r 1, r 2, ρ and a ≥ 0 satisfy the following conditions 0 < r 1 < ρ < r 2 < R , 0 < s < 1 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/math6_1.jpg"/> R + 3 A + a R + A R m + A R k < s , ρ + A < 2 B 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/math6_2.jpg"/> 3 A + a r i + A r i m + A r i k < r i , B 0 > R . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/math6_3.jpg"/> Consider the BVP 0 = Δ u + f ( x , u ) , x ∈ Ω , ∂ u ∂ n = a u + g on ∂ Ω , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/math6_4.jpg"/> where Ω ⊂ R n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/inline-math6_1.jpg"/> is a bounded open set, ∂Ω is smooth, ∫ ∂ Ω d σ ≤ A , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003028727/1d426271-33df-4258-80cb-5188d2ed19c2/content/umath6_1.jpg"/> and dσ is a vector element of ∂Ω,
