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

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).