Article Contents
Article Contents

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

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.

Mathematics Subject Classification: 34B08, 34B18, 34C23, 35Q92, 92D25.

 Citation:

• 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).

•  [1] A. Boscaggin, G. 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. [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. [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. [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. [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. [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. [7] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.  doi: 10.1007/BF00277151. [8] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [9] J. López-Gómez, M. 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. [10] J. López-Gómez, M. 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. [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. [12] J. López-Gómez, A. 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. [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. [14] Y. Lou, T. 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. [15] Y. Lou, W.-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. [16] T. Nagylaki, Conditions for the existence of clines, Genetics, 3 (1975), 595-615. [17] T. Nagylaki, L. 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. [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. [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. [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. [21] K. Nakashima, W.-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. [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. [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. [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. [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. [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.

Figures(11)