Open Access

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

Boundary Value Problems20102010:357542

DOI: 10.1155/2010/357542

Received: 13 November 2009

Accepted: 23 May 2010

Published: 21 June 2010

Abstract

We study the positive solutions to boundary value problems of the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq1_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq3_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq5_HTML.gif is a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq6_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq8_HTML.gif is the Laplace operator, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq9_HTML.gif is a positive parameter, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq10_HTML.gif is a continuous function which is sublinear at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq12_HTML.gif is the outward normal derivative, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq13_HTML.gif is a smooth function nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq14_HTML.gif . In particular, we discuss the existence of at least two positive radial solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq15_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq16_HTML.gif is an annulus in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq17_HTML.gif . Further, we discuss the existence of a double S-shaped bifurcation curve when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq19_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq20_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F357542/MediaObjects/13661_2009_Article_917_IEq21_HTML.gif .

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Authors’ Affiliations

(1)
Department of Mathematics and Statistics, Center for Computational Sciences, Mississippi State University
(2)
Department of Mathematics, Pusan National University

Copyright

© Jerome Goddard II et al. 2010

This article is published under license to BioMed Central Ltd. 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.