Alternate steady states for classes of reaction diffusion models on exterior domains

Dagny Butler; Eunkyung Ko; R. Shivaji

doi:10.3934/dcdss.2014.7.1181

Article Contents

2014, Volume 7, Issue 6: 1181-1191. Doi: 10.3934/dcdss.2014.7.1181

This issue Previous Article Connections of zero curvature and applications to nonlinear partial differential equations Next Article Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity

Alternate steady states for classes of reaction diffusion models on exterior domains

1.
Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS 39762
2.
TIFR Center for Applicable Mathematics, Yelahanka, Bangalore 560065
3.
Department of Mathematics & Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412

Received: December 31, 2012

Revised: October 31, 2013

Published: November 2014

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Abstract

We study positive radial solutions to the problem \begin{equation*} \left\{ \begin{split} -\Delta u &= \lambda K(|x|)f(u), \quad x \in \Omega, \\u(x) &= 0 \qquad

\mbox{ if } |x|=r_0, \\u(x) &\rightarrow 0 \qquad

\mbox{ as } |x|\rightarrow\infty, \end{split} \right. \end{equation*} where $\Delta u=div \big(\nabla u\big)$ is the Laplacian of $u$, $\lambda$ is a positive parameter, $\Omega=\{x\in\mathbb{R}^N: |x|>r_0\}$, $r_0>0$, and $N>2$. Here, $f\in C^2[0,\infty)$ and $f(u)>0$ on $(0,\sigma)$ and $f(u)<0$ for $u>\sigma$. Furthermore, $K:[r_0, \infty)\rightarrow(0,\infty)$ is continuous and $\lim_{r\rightarrow\infty}K(r)=0$. We discuss the existence of multiple positive solutions for a certain range of $\lambda$ leading to the occurrence of an S-shaped bifurcation curve when $f$ satisfies some additional assumptions. In particular, the two models we consider are $f_1(u)=u-\frac{u^2}{K}-c\frac{u^2}{1+u^2}$ and $f_2(u)=\tilde{K}-u+\tilde{c}\frac{u^4}{1+u^4}$. We prove our results by the method of sub-super solutions.

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Mathematics Subject Classification: Primary: 34B16, 35J60; Secondary: 92F99.

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