# American Institute of Mathematical Sciences

October  2022, 15(10): 2795-2806. doi: 10.3934/dcdss.2022067

## $\Sigma$-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 Published  October 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, 2022, 15 (10) : 2795-2806. doi: 10.3934/dcdss.2022067
##### References:
 [1] A. Acharya, N. Fonseka, J. 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. Ali, M. 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. Castro, J. 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. Cronin, J. 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. Fonseka, R. Shivaji, B. 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. Acharya, N. Fonseka, J. 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. Ali, M. 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. Castro, J. 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. Cronin, J. 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. Fonseka, R. Shivaji, B. 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.
Bifurcation diagram of (1.1) when hypotheses of Theorem 1.1(b) $(H_1-H_3)$ hold
Bifurcation diagram of (1.1) when hypotheses of Corollary 1 $(H_1-H_4)$ hold
Shape of $h$ producing multiplicity
Prototypical shapes of $f$ and $g$ producing a $\Sigma-$shaped bifurcation curve
Graph of $f$ and corresponding typical bifurcation curves for two sets of parameters $(k, \alpha)$
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