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

Abstract

<p style='text-indent:20px;'>We study the second-order boundary value problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ a_{\lambda,\mu} $\end{document}</tex-math></inline-formula> is a step-wise indefinite weight function, precisely <inline-formula><tex-math id="M2">\begin{document}$ a_{\lambda,\mu}\equiv\lambda $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ [0,\sigma]\cup[1-\sigma,1] $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ a_{\lambda,\mu}\equiv-\mu $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M5">\begin{document}$ (\sigma,1-\sigma) $\end{document}</tex-math></inline-formula>, for some <inline-formula><tex-math id="M6">\begin{document}$ \sigma\in\left(0,\frac{1}{2}\right) $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> positive real parameters. We investigate the topological structure of the set of positive solutions which lie in <inline-formula><tex-math id="M9">\begin{document}$ (0,1) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M10">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> vary. Depending on <inline-formula><tex-math id="M12">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and based on a phase-plane analysis and on time-mapping estimates, our findings lead to three different (from the topological point of view) global bifurcation diagrams of the solutions in terms of the parameter <inline-formula><tex-math id="M13">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>. Finally, for the first time in the literature, a qualitative bifurcation diagram concerning the number of solutions in the <inline-formula><tex-math id="M14">\begin{document}$ (\lambda,\mu) $\end{document}</tex-math></inline-formula>-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.</p>