$ \Sigma $-shaped bifurcation curves for classes of elliptic systems

Abstract

<p style='text-indent:20px;'>We study positive solutions to classes of steady state reaction diffusion systems of the form:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\lbrace \begin{matrix}-\Delta u = \lambda f(v) ;\; \Omega\\ -\Delta v = \lambda g(u) ;\; \Omega\\ \frac{\partial u}{\partial \eta}+\sqrt{\lambda} u = 0; \; \partial \Omega\\ \frac{\partial v}{\partial \eta}+\sqrt{\lambda}v = 0; \; \partial \Omega\ \end{matrix} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \lambda>0 $\end{document}</tex-math></inline-formula> is a positive parameter, <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M5">\begin{document}$ N > 1 $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M6">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M7">\begin{document}$ \Omega = (0, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \frac{\partial z}{\partial \eta} $\end{document}</tex-math></inline-formula> is the outward normal derivative of <inline-formula><tex-math id="M9">\begin{document}$ z $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M10">\begin{document}$ f, g \in C^2[0, r) \cap C[0, \infty) $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M11">\begin{document}$ r>0 $\end{document}</tex-math></inline-formula>. Further, we assume that <inline-formula><tex-math id="M12">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ g $\end{document}</tex-math></inline-formula> are increasing functions such that <inline-formula><tex-math id="M14">\begin{document}$ f(0) = 0 = g(0) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M15">\begin{document}$ f'(0) = g'(0) = 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M16">\begin{document}$ f''(0)>0, g''(0)>0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M17">\begin{document}$ \lim\limits_{s \rightarrow \infty} \frac{f(Mg(s))}{s} = 0 $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M18">\begin{document}$ M>0 $\end{document}</tex-math></inline-formula>. Under certain additional assumptions on <inline-formula><tex-math id="M19">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}$ g $\end{document}</tex-math></inline-formula> we prove that the bifurcation diagram for positive solutions of this system is at least <inline-formula><tex-math id="M21">\begin{document}$ \Sigma- $\end{document}</tex-math></inline-formula>shaped. We also discuss an example where <inline-formula><tex-math id="M22">\begin{document}$ f $\end{document}</tex-math></inline-formula> is sublinear at <inline-formula><tex-math id="M23">\begin{document}$ \infty $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M24">\begin{document}$ g $\end{document}</tex-math></inline-formula> is superlinear at <inline-formula><tex-math id="M25">\begin{document}$ \infty $\end{document}</tex-math></inline-formula> which satisfy our hypotheses.</p>