A Double S-Shaped Bifurcation Curve for a Reaction-Diffusion Model with Nonlinear Boundary Conditions. (21st June 2010) - Goddard Goddard, Jerome Jerome; Lee Lee, Eun Kyoung Eun Kyoung; Shivaji Shivaji, R. R.
Abstract : We study the positive solutions to boundary value problems of the form - Δ u = λ f ( u ) ; Ω, α ( x, u ) (∂ u / ∂ η) + [ 1 - α ( x, u ) ] u = 0 ; ∂ Ω, where Ω is a bounded domain in ℝ n with n ≥ 1, Δ is the Laplace operator, λ is a positive parameter, f : [ 0, ∞ ) → ( 0, ∞ ) is a continuous function which is sublinear at ∞, ∂ u / ∂ η is the outward normal derivative, and α ( x, u ) : Ω × ℝ → [ 0, 1 ] is a smooth function nondecreasing in u . In particular, we discuss the existence of at least two positive radial solutions for λ ≫ 1 when Ω is an annulus in ℝ n . Further, we discuss the existence of a double S-shaped bifurcation curve when n = 1, Ω = ( 0, 1 ), and f ( s ) = e β s / ( β + s ) with β ≫ 1 .