A Double S-Shaped Bifurcation Curve for a Reaction-Diffusion Model with Nonlinear Boundary Conditions. (21st June 2010) - Goddard, Jerome; Lee, Eun Kyoung; Shivaji, 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, whereΩ is a bounded domain inℝ n withn ≥ 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, andα ( x, u ) : Ω × ℝ → [ 0, 1 ] is a smooth function nondecreasing inu . In particular, we discuss the existence of at least two positive radial solutions forλ ≫ 1 whenΩ is an annulus inℝ n . Further, we discuss the existence of a double S-shaped bifurcation curve whenn = 1, Ω = ( 0, 1 ), andf ( s ) = e β s / ( β + s ) withβ ≫ 1 .