    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:
  H. Asakawa, Nonresonant singular two-point boundary value problems,, Nonlinear Anal., 44 (2001), 791.  doi: 10.1016/S0362-546X(99)00308-9.  Google Scholar  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  D. Butler, S. Sasi and R. Shivaji, Existence of alternate steady states in a phosphorous cycling model,, ISRN Mathematical Analysis, (2012). Google Scholar  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  E. Lee, L. Sankar and R. Shivaji, Positive solutions for infinite semipositone problems on exterior domains,, Differential Integral Equations, 24 (2011), 861. Google Scholar  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  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  E. H. Van Nes and M. Scheffer, Implications of spatial heterogeneity for catastrophic regime shifts in ecosystems,, Ecology, 86 (2005), 1797.   Google Scholar

show all references

##### References:
  H. Asakawa, Nonresonant singular two-point boundary value problems,, Nonlinear Anal., 44 (2001), 791.  doi: 10.1016/S0362-546X(99)00308-9.  Google Scholar  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  D. Butler, S. Sasi and R. Shivaji, Existence of alternate steady states in a phosphorous cycling model,, ISRN Mathematical Analysis, (2012). Google Scholar  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  E. Lee, L. Sankar and R. Shivaji, Positive solutions for infinite semipositone problems on exterior domains,, Differential Integral Equations, 24 (2011), 861. Google Scholar  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  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  E. H. Van Nes and M. Scheffer, Implications of spatial heterogeneity for catastrophic regime shifts in ecosystems,, Ecology, 86 (2005), 1797.   Google Scholar
  Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133  Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049  Heikki Haario, Leonid Kalachev, Marko Laine. Reduction and identification of dynamic models. Simple example: Generic receptor model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 417-435. doi: 10.3934/dcdsb.2013.18.417  Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216  María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic & Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024  Kimun Ryu, Inkyung Ahn. Positive steady--states for two interacting species models with linear self-cross diffusions. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1049-1061. doi: 10.3934/dcds.2003.9.1049  Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations & Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81  Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701  Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 819-833. doi: 10.3934/dcdsb.2011.16.819  Robert E. Beardmore, Rafael Peña-Miller. Antibiotic cycling versus mixing: The difficulty of using mathematical models to definitively quantify their relative merits. Mathematical Biosciences & Engineering, 2010, 7 (4) : 923-933. doi: 10.3934/mbe.2010.7.923  Manuela Caratozzolo, Santina Carnazza, Luigi Fortuna, Mattia Frasca, Salvatore Guglielmino, Giovanni Gurrieri, Giovanni Marletta. Self-organizing models of bacterial aggregation states. Mathematical Biosciences & Engineering, 2008, 5 (1) : 75-83. doi: 10.3934/mbe.2008.5.75  Youcef Amirat, Kamel Hamdache. Steady state solutions of ferrofluid flow models. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2329-2355. doi: 10.3934/cpaa.2016039  Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1  Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053  Jakub Cupera. Diffusion approximation of neuronal models revisited. Mathematical Biosciences & Engineering, 2014, 11 (1) : 11-25. doi: 10.3934/mbe.2014.11.11  Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259  Guo Lin, Wan-Tong Li, Mingju Ma. Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 393-414. doi: 10.3934/dcdsb.2010.13.393  Hiroaki Morimoto. Optimal harvesting and planting control in stochastic logistic population models. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2545-2559. doi: 10.3934/dcdsb.2012.17.2545  Urszula Ledzewicz, James Munden, Heinz Schättler. Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 415-438. doi: 10.3934/dcdsb.2009.12.415  Andrea Franceschetti, Andrea Pugliese, Dimitri Breda. Multiple endemic states in age-structured $SIR$ epidemic models. Mathematical Biosciences & Engineering, 2012, 9 (3) : 577-599. doi: 10.3934/mbe.2012.9.577

2019 Impact Factor: 1.233