Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article A Double S-Shaped Bifurcation Curve for a Reaction-Diffusion Model with Nonlinear Boundary Conditions | Hindawi Publishing Corporation Boundary Value Problems Volume 2o10 Article ID 357542 23 pages doi 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 1 Department of Mathematics and Statistics Center for Computational Sciences Mississippi State University Mississippi State MS 39762 USA 2 Department of Mathematics Pusan National University Busan 609-735 Republic of Korea Correspondence should be addressed to R. Shivaji shivaji@ Received 13 November 2009 Accepted 23 May 2010 Academic Editor Martin D. Schechter Copyright 2010 Jerome Goddard II 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. We study the positive solutions to boundary value problems of the form -Au Xf u Q. a x u du dp 1 - a x u u 0 dQ where Q is a bounded domain in Rn with n 1 A is the Laplace operator X is a positive parameter f 0 to 0 to is a continuous function which is sublinear at to du dp is the outward normal derivative and a x u Q X R 0 1 isasmooth function nondecreasing in u. In particular we discuss the existence of at least two positive radial solutions for X 1 when Q is an annulus in Rn. Further we discuss the existence of a double S-shaped bifurcation curve when n 1 Q 0 1 and f s eps p s with p 1. 1. Introduction In this paper we consider the reaction-diffusion model with nonlinear boundary condition given by ut dAu Xf u Q da x u Ệu 1 dn - a x uf u 0 dQ where Q is a bounded domain in Rn with n 1 A is the Laplace operator X is a positive parameter d is the diffusion coefficient du dn is the outward normal derivative f 0 to 0 to is a smooth function and a x ù Q X R 0 1 is a smooth function nondecreasing 2 Boundary Value Problems in u. The boundary condition .
