Boundary Value Problems
Volume 2010 (2010), Article ID 357542, 23 pages
doi:10.1155/2010/357542
Research Article

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

Jerome Goddard II,1 Eun Kyoung Lee,2 and R. Shivaji1

1Department of Mathematics and Statistics, Center for Computational Sciences, Mississippi State University, Mississippi State, MS 39762, USA
2Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea

Received 13 November 2009; Accepted 23 May 2010

Academic Editor: Martin D. Schechter

Copyright © 2010 Jerome Goddard et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 ℝn 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 ℝn. Further, we discuss the existence of a double S-shaped bifurcation curve when n=1, Ω=(0,1), and f(s)=eβs/(β+s) with β≫1.