# American Institute of Mathematical Sciences

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
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##### References:
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).
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
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)
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 $P(s) = (u_{s}(\sigma),v_{s}(\sigma))$
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)
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)
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 $P(s) = (u_{s}(\sigma),v_{s}(\sigma))$
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 $\tilde{\mu}(\lambda)$
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 $\mathfrak{B}_{+}$
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
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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