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. 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) $
[1]

Julián López-Góme, Andrea Tellini, F. Zanolin. High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Communications on Pure and Applied Analysis, 2014, 13 (1) : 1-73. doi: 10.3934/cpaa.2014.13.1

[2]

Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3443-3462. doi: 10.3934/dcds.2019142

[3]

Xuemei Zhang, Meiqiang Feng. Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2149-2171. doi: 10.3934/cpaa.2018103

[4]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061

[5]

Jean Dolbeault, Robert Stańczy. Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 139-154. doi: 10.3934/dcds.2015.35.139

[6]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147

[7]

Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047

[8]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107

[9]

Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044

[10]

Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012

[11]

Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911

[12]

Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080

[13]

Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166

[14]

Lynnyngs Kelly Arruda, Francisco Odair de Paiva, Ilma Marques. A remark on multiplicity of positive solutions for a class of quasilinear elliptic systems. Conference Publications, 2011, 2011 (Special) : 112-116. doi: 10.3934/proc.2011.2011.112

[15]

Masataka Shibata. Multiplicity of positive solutions to semi-linear elliptic problems on metric graphs. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4107-4126. doi: 10.3934/cpaa.2021147

[16]

Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105

[17]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Spiraling bifurcation diagrams in superlinear indefinite problems. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1561-1588. doi: 10.3934/dcds.2015.35.1561

[18]

E. Kapsza, Gy. Károlyi, S. Kovács, G. Domokos. Regular and random patterns in complex bifurcation diagrams. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 519-540. doi: 10.3934/dcdsb.2003.3.519

[19]

Po-Chun Huang, Shin-Hwa Wang, Tzung-Shin Yeh. Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297

[20]

Kuan-Ju Huang, Yi-Jung Lee, Tzung-Shin Yeh. Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1497-1514. doi: 10.3934/cpaa.2016.15.1497

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (194)
  • HTML views (92)
  • Cited by (0)

Other articles
by authors

[Back to Top]