Σ-shaped bifurcation curves influenced by nonlinear boundary conditions for classes of reaction diffusion systems

Abstract

We analyse positive solutions to steady state reaction diffusion systems of the form: \begin{equation*} \begin{cases} - \Delta u = \lambda f_1(v) &\text{in } \Omega, \\ - \Delta v = \lambda f_2(u)&\text{in } \Omega, \\ \noalign{\vskip3pt} \dfrac{\partial u}{\partial \eta}+ \sqrt{\lambda}g_1(v)u = 0 &\text{in }\partial \Omega,\\ \noalign{\vskip3pt} \dfrac{\partial v}{\partial \eta}+ \sqrt{\lambda}g_2(u)v = 0 &\text{in }\partial \Omega, \end{cases} \end{equation*} where $\lambda> 0$ is a 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 $z$, $f_1, f_2 \in C([0, \infty) , [0, \infty))$ are increasing functions, differentiable on $[0, r)$ for some $r> 0$, $f_1(0) = f_2(0) = 0$, $f_1'(0) = f_2'(0) = 1$, $ \lim\limits_{s \to \infty} {f_1(Mf_2(s))}/{s} = 0 $ for each $M> 0$ ($f_1, f_2$ satisfy a combined sublinear condition at infinity), $g_1, g_2 \in C([0, \infty) , (0, 1])$ are nonincreasing functions such that $g_1(0)= g_2(0)=1$, and $\underline{g} := \min\Big \{\lim\limits_{s \rightarrow \infty} g_1(s), \lim\limits_{s \rightarrow \infty} g_2(s)\Big\} > 0$. We discuss the existence of multiple positive solutions for certain ranges of $\lambda$ leading to the occurrence of $\Sigma$-shaped bifurcation diagrams. Our results are established via the method of sub-supersolutions.