$ \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{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*} $

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