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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
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  • 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:

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  • 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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