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

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a class of steady state reaction diffusion systems with multiple parameters, using sub-super solution methods under certain conditions.

Contribution

It provides new existence and multiplicity results for positive solutions of reaction diffusion systems with parameter-dependent boundary conditions, extending previous work.

Findings

Existence of positive solutions under specific conditions.

Multiple positive solutions can occur depending on parameters.

Results apply to systems with nonlinearities satisfying growth conditions.

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.