American Institute of Mathematical Sciences

December  2014, 7(6): 1181-1191. doi: 10.3934/dcdss.2014.7.1181

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

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

Received  January 2013 Revised  November 2013 Published  June 2014

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.
Citation: Dagny Butler, Eunkyung Ko, R. Shivaji. Alternate steady states for classes of reaction diffusion models on exterior domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1181-1191. doi: 10.3934/dcdss.2014.7.1181
References:
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References:
 [1] H. Asakawa, Nonresonant singular two-point boundary value problems,, Nonlinear Anal., 44 (2001), 791. doi: 10.1016/S0362-546X(99)00308-9. Google Scholar [2] A. K. Ben-Naoum and C. D. Coster, On the existence and multiplicity of positive solutions of the p-Laplacian separated boundary value problem,, Differential and Integral Equations, 10 (1997), 1093. Google Scholar [3] D. Butler, S. Sasi and R. Shivaji, Existence of alternate steady states in a phosphorous cycling model,, ISRN Mathematical Analysis, (2012). Google Scholar [4] S. R. Carpenter, D. Ludwig and W. A. Brock, Management of eutrophication for lakes subject to potentially irreversible change,, Ecological Applications, 9 (1999), 751. Google Scholar [5] E. Lee, L. Sankar and R. Shivaji, Positive solutions for infinite semipositone problems on exterior domains,, Differential Integral Equations, 24 (2011), 861. Google Scholar [6] E. Lee, S. Sasi and R. Shivaji, S-shaped bifurcation curves in ecosystems,, J. Math. Anal. Appl., 381 (2011), 732. doi: 10.1016/j.jmaa.2011.03.048. Google Scholar [7] M. Scheffer, W. Brock and F. Westley, Socioeconomic mechanisms preventing optimum use of ecosystem services: An interdisciplinary theoretical analysis,, Ecosystems, 3 (2000), 451. doi: 10.1007/s100210000040. Google Scholar [8] E. H. Van Nes and M. Scheffer, Implications of spatial heterogeneity for catastrophic regime shifts in ecosystems,, Ecology, 86 (2005), 1797. Google Scholar
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