January  2022, 42(1): 21-71. doi: 10.3934/dcds.2021107

On the number of positive solutions to an indefinite parameter-dependent Neumann problem

1. 

Department of Mathematics, Computer Science and Physics, University of Udine, Via delle Scienze 206, 33100 Udine, Italy

2. 

École des Hautes Études en Sciences Sociales, Centre d'Analyse et de Mathématique Sociales (CAMS), CNRS, 54 Boulevard Raspail, 75006 Paris, France

3. 

Department of Applied Mathematics in Industrial Engineering, E.T.S.I.D.I., Universidad Politécnica de Madrid, Ronda de Valencia 3, 28012 Madrid, Spain

* Corresponding author: guglielmo.feltrin@uniud.it

Received  January 2021 Revised  May 2021 Published  January 2022 Early access  July 2021

Fund Project: Work written under the auspices of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author has been supported by the Fondation Sciences Mathématiques de Paris (FSMP) through the project: "Reaction-Diffusion Equations in Population Genetics: a study of the influence of geographical barriers on traveling waves and non-constant stationary solutions''. The third author has received funding from project PGC2018-097104-B-100 of the Spanish Ministry of Science, Innovation and Universities and from the Escuela Técnica Superior de Ingeniería y Diseño Industrial of the Universidad Politécnica de Madrid

We study the second-order boundary value problem
$ \begin{equation*} \begin{cases}\, -u'' = a_{\lambda,\mu}(t) \, u^{2}(1-u), & t\in(0,1), \\\, u'(0) = 0, \quad u'(1) = 0,\end{cases} \end{equation*} $
where
$ a_{\lambda,\mu} $
is a step-wise indefinite weight function, precisely
$ a_{\lambda,\mu}\equiv\lambda $
in
$ [0,\sigma]\cup[1-\sigma,1] $
and
$ a_{\lambda,\mu}\equiv-\mu $
in
$ (\sigma,1-\sigma) $
, for some
$ \sigma\in\left(0,\frac{1}{2}\right) $
, with
$ \lambda $
and
$ \mu $
positive real parameters. We investigate the topological structure of the set of positive solutions which lie in
$ (0,1) $
as
$ \lambda $
and
$ \mu $
vary. Depending on
$ \lambda $
and based on a phase-plane analysis and on time-mapping estimates, our findings lead to three different (from the topological point of view) global bifurcation diagrams of the solutions in terms of the parameter
$ \mu $
. Finally, for the first time in the literature, a qualitative bifurcation diagram concerning the number of solutions in the
$ (\lambda,\mu) $
-plane is depicted. The analyzed Neumann problem has an application in the analysis of stationary solutions to reaction-diffusion equations in population genetics driven by migration and selection.
Citation: Guglielmo Feltrin, Elisa Sovrano, Andrea Tellini. On the number of positive solutions to an indefinite parameter-dependent Neumann problem. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 21-71. doi: 10.3934/dcds.2021107
References:
[1]

A. BoscagginG. Feltrin and E. Sovrano, High multiplicity and chaos for an indefinite problem arising from genetic models, Adv. Nonlinear Stud., 20 (2020), 675-699.  doi: 10.1515/ans-2020-2094.  Google Scholar

[2]

K. J. Brown and P. Hess, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Differential Integral Equations, 3 (1990), 201-207.   Google Scholar

[3]

G. Feltrin, Positive Solutions to Indefinite Problems, A Topological Approach, Frontiers in Mathematics, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-94238-4.  Google Scholar

[4]

G. Feltrin and P. Gidoni, Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model, Nonlinear Anal. Real World Appl., 54 (2020), 103108. doi: 10.1016/j.nonrwa.2020.103108.  Google Scholar

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G. Feltrin and E. Sovrano, An indefinite nonlinear problem in population dynamics: High multiplicity of positive solutions, Nonlinearity, 31 (2018), 4137-4161.  doi: 10.1088/1361-6544/aac8bb.  Google Scholar

[6]

G. Feltrin and E. Sovrano, Three positive solutions to an indefinite Neumann problem: A shooting method, Nonlinear Anal., 166 (2018), 87-101.  doi: 10.1016/j.na.2017.10.006.  Google Scholar

[7]

W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.  doi: 10.1007/BF00277151.  Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

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J. López-GómezM. Molina-Meyer and A. Tellini, The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions, J. Differential Equations, 255 (2013), 503-523.  doi: 10.1016/j.jde.2013.04.019.  Google Scholar

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J. López-GómezM. Molina-Meyer and A. Tellini, Intricate dynamics caused by facilitation in competitive environments within polluted habitat patches, European J. Appl. Math., 25 (2014), 213-229.  doi: 10.1017/S0956792513000429.  Google Scholar

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J. López-Gómez and A. Tellini, Generating an arbitrarily large number of isolas in a superlinear indefinite problem, Nonlinear Anal., 108 (2014), 223-248.  doi: 10.1016/j.na.2014.06.003.  Google Scholar

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J. López-GómezA. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems, Commun. Pure Appl. Anal., 13 (2014), 1-73.  doi: 10.3934/cpaa.2014.13.1.  Google Scholar

[13]

Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics, J. Differential Equations, 181 (2002), 388-418.  doi: 10.1006/jdeq.2001.4086.  Google Scholar

[14]

Y. LouT. Nagylaki and W.-M. Ni, An introduction to migration-selection PDE models, Discrete Contin. Dyn. Syst., 33 (2013), 4349-4373.  doi: 10.3934/dcds.2013.33.4349.  Google Scholar

[15]

Y. LouW.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics. II. Stability and multiplicity, Discrete Contin. Dyn. Syst., 27 (2010), 643-655.  doi: 10.3934/dcds.2010.27.643.  Google Scholar

[16]

T. Nagylaki, Conditions for the existence of clines, Genetics, 3 (1975), 595-615.   Google Scholar

[17]

T. NagylakiL. Su and T. F. Dupont, Uniqueness and multiplicity of clines in an environmental pocket, Theoretical Population Biology, 130 (2019), 106-131.  doi: 10.1016/j.tpb.2019.07.006.  Google Scholar

[18]

K. Nakashima, The uniqueness of indefinite nonlinear diffusion problem in population genetics, part I, J. Differential Equations, 261 (2016), 6233-6282.  doi: 10.1016/j.jde.2016.08.041.  Google Scholar

[19]

K. Nakashima, The uniqueness of an indefinite nonlinear diffusion problem in population genetics, part II, J. Differential Equations, 264 (2018), 1946-1983.  doi: 10.1016/j.jde.2017.10.014.  Google Scholar

[20]

K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, 268 (2020), 7803-7842.  doi: 10.1016/j.jde.2019.11.082.  Google Scholar

[21]

K. NakashimaW.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics. I. Existence and limiting profiles, Discrete Contin. Dyn. Syst., 27 (2010), 617-641.  doi: 10.3934/dcds.2010.27.617.  Google Scholar

[22]

K. Nakashima and L. Su, Nonuniqueness of an indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, 269 (2020), 4643-4682.  doi: 10.1016/j.jde.2020.03.042.  Google Scholar

[23]

P. Omari and E. Sovrano, Positive solutions of indefinite logistic growth models with flux-saturated diffusion, Nonlinear Anal., 201 (2020), 111949. doi: 10.1016/j.na.2020.111949.  Google Scholar

[24]

E. Sovrano, A negative answer to a conjecture arising in the study of selection-migration models in population genetics, J. Math. Biol., 76 (2018), 1655-1672.  doi: 10.1007/s00285-017-1185-7.  Google Scholar

[25]

A. Tellini, Imperfect bifurcations via topological methods in superlinear indefinite problems, Discrete Contin. Dyn. Syst. (Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl.), (2015), 1050–1059. doi: 10.3934/proc.2015.1050.  Google Scholar

[26]

A. Tellini, High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions, J. Math. Anal. Appl., 467 (2018), 673-698.  doi: 10.1016/j.jmaa.2018.07.034.  Google Scholar

show all references

References:
[1]

A. BoscagginG. Feltrin and E. Sovrano, High multiplicity and chaos for an indefinite problem arising from genetic models, Adv. Nonlinear Stud., 20 (2020), 675-699.  doi: 10.1515/ans-2020-2094.  Google Scholar

[2]

K. J. Brown and P. Hess, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Differential Integral Equations, 3 (1990), 201-207.   Google Scholar

[3]

G. Feltrin, Positive Solutions to Indefinite Problems, A Topological Approach, Frontiers in Mathematics, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-94238-4.  Google Scholar

[4]

G. Feltrin and P. Gidoni, Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model, Nonlinear Anal. Real World Appl., 54 (2020), 103108. doi: 10.1016/j.nonrwa.2020.103108.  Google Scholar

[5]

G. Feltrin and E. Sovrano, An indefinite nonlinear problem in population dynamics: High multiplicity of positive solutions, Nonlinearity, 31 (2018), 4137-4161.  doi: 10.1088/1361-6544/aac8bb.  Google Scholar

[6]

G. Feltrin and E. Sovrano, Three positive solutions to an indefinite Neumann problem: A shooting method, Nonlinear Anal., 166 (2018), 87-101.  doi: 10.1016/j.na.2017.10.006.  Google Scholar

[7]

W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.  doi: 10.1007/BF00277151.  Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[9]

J. López-GómezM. Molina-Meyer and A. Tellini, The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions, J. Differential Equations, 255 (2013), 503-523.  doi: 10.1016/j.jde.2013.04.019.  Google Scholar

[10]

J. López-GómezM. Molina-Meyer and A. Tellini, Intricate dynamics caused by facilitation in competitive environments within polluted habitat patches, European J. Appl. Math., 25 (2014), 213-229.  doi: 10.1017/S0956792513000429.  Google Scholar

[11]

J. López-Gómez and A. Tellini, Generating an arbitrarily large number of isolas in a superlinear indefinite problem, Nonlinear Anal., 108 (2014), 223-248.  doi: 10.1016/j.na.2014.06.003.  Google Scholar

[12]

J. López-GómezA. Tellini and F. Zanolin, High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems, Commun. Pure Appl. Anal., 13 (2014), 1-73.  doi: 10.3934/cpaa.2014.13.1.  Google Scholar

[13]

Y. Lou and T. Nagylaki, A semilinear parabolic system for migration and selection in population genetics, J. Differential Equations, 181 (2002), 388-418.  doi: 10.1006/jdeq.2001.4086.  Google Scholar

[14]

Y. LouT. Nagylaki and W.-M. Ni, An introduction to migration-selection PDE models, Discrete Contin. Dyn. Syst., 33 (2013), 4349-4373.  doi: 10.3934/dcds.2013.33.4349.  Google Scholar

[15]

Y. LouW.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics. II. Stability and multiplicity, Discrete Contin. Dyn. Syst., 27 (2010), 643-655.  doi: 10.3934/dcds.2010.27.643.  Google Scholar

[16]

T. Nagylaki, Conditions for the existence of clines, Genetics, 3 (1975), 595-615.   Google Scholar

[17]

T. NagylakiL. Su and T. F. Dupont, Uniqueness and multiplicity of clines in an environmental pocket, Theoretical Population Biology, 130 (2019), 106-131.  doi: 10.1016/j.tpb.2019.07.006.  Google Scholar

[18]

K. Nakashima, The uniqueness of indefinite nonlinear diffusion problem in population genetics, part I, J. Differential Equations, 261 (2016), 6233-6282.  doi: 10.1016/j.jde.2016.08.041.  Google Scholar

[19]

K. Nakashima, The uniqueness of an indefinite nonlinear diffusion problem in population genetics, part II, J. Differential Equations, 264 (2018), 1946-1983.  doi: 10.1016/j.jde.2017.10.014.  Google Scholar

[20]

K. Nakashima, Multiple existence of indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, 268 (2020), 7803-7842.  doi: 10.1016/j.jde.2019.11.082.  Google Scholar

[21]

K. NakashimaW.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics. I. Existence and limiting profiles, Discrete Contin. Dyn. Syst., 27 (2010), 617-641.  doi: 10.3934/dcds.2010.27.617.  Google Scholar

[22]

K. Nakashima and L. Su, Nonuniqueness of an indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, 269 (2020), 4643-4682.  doi: 10.1016/j.jde.2020.03.042.  Google Scholar

[23]

P. Omari and E. Sovrano, Positive solutions of indefinite logistic growth models with flux-saturated diffusion, Nonlinear Anal., 201 (2020), 111949. doi: 10.1016/j.na.2020.111949.  Google Scholar

[24]

E. Sovrano, A negative answer to a conjecture arising in the study of selection-migration models in population genetics, J. Math. Biol., 76 (2018), 1655-1672.  doi: 10.1007/s00285-017-1185-7.  Google Scholar

[25]

A. Tellini, Imperfect bifurcations via topological methods in superlinear indefinite problems, Discrete Contin. Dyn. Syst. (Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl.), (2015), 1050–1059. doi: 10.3934/proc.2015.1050.  Google Scholar

[26]

A. Tellini, High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions, J. Math. Anal. Appl., 467 (2018), 673-698.  doi: 10.1016/j.jmaa.2018.07.034.  Google Scholar

Figure 1.  Qualitative bifurcation digram of the solutions to (1.1) in the $ (\lambda,\mu) $-plane. The curves $ \mu^{*}_0(\lambda) $ and $ \mu^{**}_0(\lambda) $ define the non-existence region (gray). The curves $ \mu^{*}_1(\lambda) $ and $ \mu^{**}_2(\lambda) $ mark out regions of existence: at least one solution in between $ \mu^{*}_{1}(\lambda) $ and $ \mu^{*}_{2}(\lambda) $ (red) and at least two solutions in between $ \mu^*_2(\lambda) $ and $ \mu^{**}_{2}(\lambda) $ (blue). For $ \lambda\in[\lambda^{*},+\infty) $, at least four solutions in the region above $ \mu^*_4(\lambda) $ (yellow), and at least eight solutions in the one above $ \mu^*_8(\lambda) $ (green).
Figure 2.  Qualitative graph of the function $ T_{0} $ when $ \lambda\in(0,\lambda^{*}) $ (left), $ \lambda = \lambda^{*} $ (center), and $ \lambda>\lambda^{*} $ (right). The points $ s^{*} $, $ s_{0} $, and $ s_{1} $ are defined in Proposition 2.1
Figure 3.  Qualitative representation in the $ (u,v) $-plane of the curve $ \Gamma_{0}(\lambda) $ defined in (2.11) when $ \lambda\in(0,\lambda^{*}) $ (left), $ \lambda = \lambda^{*} $ (center), and $ \lambda>\lambda^{*} $ (right)
Figure 4.  For $ \lambda\in[\lambda^*,+\infty) $, qualitative representations of the graph of the function $ h_{\mu} $ defined in (4.4) (left) and of $ \Gamma_0 $ (blue), $ \Gamma_1 $ (violet), $ \mathcal{M}_{\mu} $ (pink) along with some level lines of (3.2) (gray) in the $ (u,v) $-plane (right). We set $ P(s) = (u_{s}(\sigma),v_{s}(\sigma)) $
Figure 5.  For $ \lambda\in[\lambda^*,+\infty) $, qualitative graphs of $ T_{i} $, with $ i = 1,2,3 $, which are the times necessary to connect $ \Gamma_{0} $ to $ \Gamma_{1} $, as functions of the initial data $ u(0) = s $ of (2.5): $ T_1 $ (green), $ T_2 $ (blue), and $ T_3 $ (pink)
Figure 6.  For $ \lambda\in(0,\lambda^{*}) $ and $ \mu>\tilde{\mu}(\lambda) $, qualitative graph of $ T_{1} $, which is the time necessary to connect $ \Gamma_{0} $ to $ \Gamma_{1} $, as a function of the initial data $ u(0) = s $ of (2.5)
Figure 7.  For $ \lambda\in(0,\lambda^{*}) $, qualitative representations of the graph of the function $ h_{\mu} $ defined in (4.4) (left) and of $ \Gamma_0 $ (blue), $ \Gamma_1 $ (violet), $ \mathcal{M}_{\mu} $ (pink) along with some level lines of (3.2) (gray) in the $ (u,v) $-plane (right). We set $ P(s) = (u_{s}(\sigma),v_{s}(\sigma)) $
Figure 8.  For $ \lambda\in(0,\lambda^{*}) $ and $ \mu>\tilde{\mu}(\lambda) $, qualitative graphs of $ T_{i} $, with $ i = 1,2,3 $, which are the times necessary to connect $ \Gamma_{0} $ to $ \Gamma_{1} $, as functions of the initial data $ u(0) = s $ of (2.5): $ T_1 $ (green), $ T_2 $ (blue), and $ T_3 $ (pink). The existence of the loop is ensured only for $ \mu $ near $ \tilde{\mu}(\lambda) $
Figure 9.  For $ \lambda\in(0,\lambda^{*}) $, qualitative representation in the $ (u,v) $-plane of the curves $ \Gamma_0 $ and $ \Gamma_1 $, the manifold $ \mathcal{M}_{\mu} $, and the curves $ \mathfrak{B}_{-} $ and $ \mathfrak{B}_{+} $
Figure 10.  For $ \lambda\in[\lambda^*,+\infty) $ fixed, a minimal qualitative bifurcation diagram for (1.1) with $ \mu $ as bifurcation parameter. The topological configuration involves: two unbounded branches bifurcating from $ 0 $ and $ 1 $ (black) and three unbounded branches (yellow, green) originating from a supercritical pitchfork bifurcation and two supercritical turning points, respectively
Figure 11.  For $ \lambda\in(0,\lambda^*) $ fixed, minimal qualitative bifurcation diagrams of (1.1) with $ \mu $ as bifurcation parameter. Depending on $ \lambda $, different topological configurations may appear: only a bounded connected branch (black) connecting $ 0 $ to $ 1 $ producing one to two solutions (left) or a connected branch crossed by a loop (magenta) which increases the number of solutions up to either four (center) or eight (right).
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