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

We study positive solutions to classes of steady state reaction diffusion systems of the form:\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*} $\end{document}where \begin{document}$ \lambda>0 $\end{document} is a positive parameter, \begin{document}$ \Omega $\end{document} is a bounded domain in \begin{document}$ \mathbb{R}^N $\end{document} ; \begin{document}$ N > 1 $\end{document} with smooth boundary \begin{document}$ \partial \Omega $\end{document} or \begin{document}$ \Omega = (0, 1) $\end{document} , \begin{document}$ \frac{\partial z}{\partial \eta} $\end{document} is the outward normal derivative of \begin{document}$ z $\end{document} . Here \begin{document}$ f, g \in C^2[0, r) \cap C[0, \infty) $\end{document} for some \begin{document}$ r>0 $\end{document} . Further, we assume that \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document} are increasing functions such that \begin{document}$ f(0) = 0 = g(0) $\end{document} , \begin{document}$ f'(0) = g'(0) = 1 $\end{document} , \begin{document}$ f''(0)>0, g''(0)>0 $\end{document} , and \begin{document}$ \lim\limits_{s \rightarrow \infty} \frac{f(Mg(s))}{s} = 0 $\end{document} for all \begin{document}$ M>0 $\end{document} . Under certain additional assumptions on \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document} we prove that the bifurcation diagram for positive solutions of this system is at least \begin{document}$ \Sigma- $\end{document} shaped. We also discuss an example where \begin{document}$ f $\end{document} is sublinear at \begin{document}$ \infty $\end{document} and \begin{document}$ g $\end{document} is superlinear at \begin{document}$ \infty $\end{document} which satisfy our hypotheses.