On the existence and multiplicity of positive solutions to classes of steady state reaction diffusion systems with multiple parameters (2307.12000v1)

Abstract: We study positive solutions to the steady state reaction diffusion systems of the form: \begin{equation} \left{\begin{array}{ll} -\Delta u = \lambda f(v)+\mu h(u), & \Omega,\ -\Delta v = \lambda g(u)+\mu q(v),& \Omega,\ \frac{\partial u}{\partial \eta}+\sqrt[]{\lambda +\mu}\, u=0,& \partial\Omega,\ \frac{\partial v}{\partial \eta}+\sqrt[]{\lambda +\mu}\, v=0, & \partial\Omega,\ \end{array}\right. \end{equation} where ${\lambda,\mu>0}$ are positive parameters, ${\Omega}$ is a bounded 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 $f, g, h, q\in C^{2} [0,r)\cap C[0,\infty)$ for some $r>0$. Further, we assume that $f, g, h$ and $q$ are increasing functions such that $f(0) = g(0) = h(0) = {q}(0) = 0$, $f^\prime(0), g^\prime(0), h^\prime(0), q^\prime(0) > 0$, and $\lim\limits_{s\to \infty}\frac{f(M g(s))}{s}=0$ for all $M>0$. Under certain additional assumptions on $f, g, h$ and $ q$ we prove our existence and multiplicity results. Our existence and multiplicity results are proved using sub-super solution methods.