A Double S-Shaped Bifurcation Curve for a Reaction-Diffusion Model with Nonlinear Boundary Conditions

We study the positive solutions to boundary value problems of the form −Δu λf u ; Ω, α x, u ∂u/∂η 1 − α x, u u 0; ∂Ω, where Ω is a bounded domain in R 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. Further, we discuss the existence of a double S-shaped bifurcation curve when n 1, Ω 0, 1 , and f s e β s with β 1.