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

作者：Goddard Jerome II; Lee Eun Kyoung; Shivaji R^*

来源：Boundary Value Problems, 2010, 357542.

DOI：10.1155/2010/357542

We study the positive solutions to boundary value problems of the form -Delta u = lambda f (u); Omega, alpha(x, u) (partial derivative u/partial derivative eta) + [1 - alpha(x, u)]u = 0; partial derivative Omega, where Omega is a bounded domain in R(n) with n >= 1, Delta is the Laplace operator, lambda is a positive parameter, f : [0, infinity) -> (0, infinity) is a continuous function which is sublinear at infinity, partial derivative u/partial derivative eta is the outward normal derivative, and alpha (x, u) : Omega x R -> [0, 1] is a smooth function nondecreasing in u. In particular, we discuss the existence of at least two positive radial solutions for lambda >> 1 when Omega is an annulus in R(n). Further, we discuss the existence of a double S-shaped bifurcation curve when n = 1, Omega = (0, 1), and f (s) = e(beta s/(beta+s)) with beta >> 1.

• 出版日期2010