doi: 10.3934/dcdss.2022067
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$ \Sigma $-shaped bifurcation curves for classes of elliptic systems

1. 

The University of North Carolina at Greensboro, PO Box 26170, Greensboro, NC 27402-6170, USA

2. 

School of Arts and Sciences, Carolina University, Winston-Salem, NC 27101, USA

* Corresponding author: r_shivaj@uncg.edu

Received  July 2021 Revised  January 2022 Early access March 2022

We study positive solutions to classes of steady state reaction diffusion systems of the form:
$ \begin{equation*} \left\lbrace \begin{matrix}-\Delta u = \lambda f(v) ;\; \Omega\\ -\Delta v = \lambda g(u) ;\; \Omega\\ \frac{\partial u}{\partial \eta}+\sqrt{\lambda} u = 0; \; \partial \Omega\\ \frac{\partial v}{\partial \eta}+\sqrt{\lambda}v = 0; \; \partial \Omega\ \end{matrix} \right. \end{equation*} $
where
$ \lambda>0 $
is a positive parameter,
$ \Omega $
is a bounded domain in
$ \mathbb{R}^N $
;
$ N > 1 $
with smooth boundary
$ \partial \Omega $
or
$ \Omega = (0, 1) $
,
$ \frac{\partial z}{\partial \eta} $
is the outward normal derivative of
$ z $
. Here
$ f, g \in C^2[0, r) \cap C[0, \infty) $
for some
$ r>0 $
. Further, we assume that
$ f $
and
$ g $
are increasing functions such that
$ f(0) = 0 = g(0) $
,
$ f'(0) = g'(0) = 1 $
,
$ f''(0)>0, g''(0)>0 $
, and
$ \lim\limits_{s \rightarrow \infty} \frac{f(Mg(s))}{s} = 0 $
for all
$ M>0 $
. Under certain additional assumptions on
$ f $
and
$ g $
we prove that the bifurcation diagram for positive solutions of this system is at least
$ \Sigma- $
shaped. We also discuss an example where
$ f $
is sublinear at
$ \infty $
and
$ g $
is superlinear at
$ \infty $
which satisfy our hypotheses.
Citation: Ananta Acharya, R. Shivaji, Nalin Fonseka. $ \Sigma $-shaped bifurcation curves for classes of elliptic systems. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022067
References:
[1]

A. AcharyaN. FonsekaJ. Quiroa and R. Shivaji, $\Sigma$-Shaped Bifurcation Curves, Adv. Nonlinear Anal., 10 (2021), 1255-1266.  doi: 10.1515/anona-2020-0180.

[2]

A. Acharya, N. Fonseka and R. Shivaji, Analysis of reaction diffusion systems where a parameter influences both the reaction terms as well as the bounday, Bound. Value Probl., (2021), Paper No. 15, 8 pp. doi: 10.1186/s13661-021-01490-0.

[3]

J. AliM. Ramaswamy and R. Shivaji, Multiple positive solutions for classes of elliptic systems with combined nonlinear effects, Differential Integral Equations, 19 (2006), 669-680. 

[4]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.

[5]

A. CastroJ. B. Garner and R. Shivaji, Existence results for classes of sub-linear semipositone problems, Results Math., 23 (1993), 214-220.  doi: 10.1007/BF03322297.

[6]

J. T. CroninJ. Goddard and R. Shivaji, Effects of patch-matrix composition and individual movement response on population persistence at the patch-level, Bull. Math. Biol., 81 (2019), 3933-3975.  doi: 10.1007/s11538-019-00634-9.

[7]

N. Fonseka, J. Machado and R. Shivaji, A study of logistic growth models influenced by the exterior matrix hostility and Grazing in an interior patch, Electron J. Qual. Theory Differ. Equ., (2020), Paper No. 17, 11 pp. doi: 10.14232/ejqtde.2020.1.17.

[8]

N. FonsekaR. ShivajiB. Son and K. Spetzer, Classes of reaction diffusion equations where a parameter influences the equation as well as the boundary condition, J. Math. Anal. Appl., 476 (2019), 480-494.  doi: 10.1016/j.jmaa.2019.03.053.

[9]

J. Goddard II, Q. A. Morris, S. B. Robinson and R. Shivaji, An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics, Bound. Value Probl., (2018), Paper No. 170, 17 pp. doi: 10.1186/s13661-018-1090-z.

[10]

R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications (Arlington, Tex., 1986), Lecture Notes in Pure and Appl. Math., Dekker, New York, 109 (1987), 561-566. 

show all references

References:
[1]

A. AcharyaN. FonsekaJ. Quiroa and R. Shivaji, $\Sigma$-Shaped Bifurcation Curves, Adv. Nonlinear Anal., 10 (2021), 1255-1266.  doi: 10.1515/anona-2020-0180.

[2]

A. Acharya, N. Fonseka and R. Shivaji, Analysis of reaction diffusion systems where a parameter influences both the reaction terms as well as the bounday, Bound. Value Probl., (2021), Paper No. 15, 8 pp. doi: 10.1186/s13661-021-01490-0.

[3]

J. AliM. Ramaswamy and R. Shivaji, Multiple positive solutions for classes of elliptic systems with combined nonlinear effects, Differential Integral Equations, 19 (2006), 669-680. 

[4]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.

[5]

A. CastroJ. B. Garner and R. Shivaji, Existence results for classes of sub-linear semipositone problems, Results Math., 23 (1993), 214-220.  doi: 10.1007/BF03322297.

[6]

J. T. CroninJ. Goddard and R. Shivaji, Effects of patch-matrix composition and individual movement response on population persistence at the patch-level, Bull. Math. Biol., 81 (2019), 3933-3975.  doi: 10.1007/s11538-019-00634-9.

[7]

N. Fonseka, J. Machado and R. Shivaji, A study of logistic growth models influenced by the exterior matrix hostility and Grazing in an interior patch, Electron J. Qual. Theory Differ. Equ., (2020), Paper No. 17, 11 pp. doi: 10.14232/ejqtde.2020.1.17.

[8]

N. FonsekaR. ShivajiB. Son and K. Spetzer, Classes of reaction diffusion equations where a parameter influences the equation as well as the boundary condition, J. Math. Anal. Appl., 476 (2019), 480-494.  doi: 10.1016/j.jmaa.2019.03.053.

[9]

J. Goddard II, Q. A. Morris, S. B. Robinson and R. Shivaji, An exact bifurcation diagram for a reaction diffusion equation arising in population dynamics, Bound. Value Probl., (2018), Paper No. 170, 17 pp. doi: 10.1186/s13661-018-1090-z.

[10]

R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Applications (Arlington, Tex., 1986), Lecture Notes in Pure and Appl. Math., Dekker, New York, 109 (1987), 561-566. 

Figure 1.  Bifurcation diagram of (1.1) when hypotheses of Theorem 1.1(b) $ (H_1-H_3) $ hold
Figure 2.  Bifurcation diagram of (1.1) when hypotheses of Corollary 1 $ (H_1-H_4) $ hold
Figure 3.  Shape of $ h $ producing multiplicity
Figure 4.  Prototypical shapes of $ f $ and $ g $ producing a $ \Sigma- $shaped bifurcation curve
Figure 5.  Graph of $ f $ and corresponding typical bifurcation curves for two sets of parameters $ (k, \alpha) $
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