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

We study positive solutions to the steady state reaction diffusion systems of the form: \begin{equation} \left{\begin{array}{ll} -Δu = λf(v)+μh(u), & Ω,\ -Δv = λg(u)+μq(v),& Ω,\ \frac{\partial u}{\partial η}+\sqrt[]{λ+μ}, u=0,& \partialΩ,\ \frac{\partial v}{\partial η}+\sqrt[]{λ+μ}, v=0, & \partialΩ,\ \end{array}\right. \end{equation} where $\lambda ,\mu >0$ are positive parameters, $\mathrm{\Omega }$ is a bounded in ${\mathbb{R}}^N$$(N>1)$ with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$, or $\mathrm{\Omega }=(0,1)$, $\frac{\mathrm{\partial }z}{\mathrm{\partial }\eta }$ is the outward normal derivative of $z$. Here $f,g,h,q\in C^2[0,r)\cap C[0,\mathrm{\infty })$ for some $r>0$. Further, we assume that $f,g,h$ and $q$ are increasing functions such that $f(0)=g(0)=h(0)=q(0)=0$, $f^{\mathrm{\prime }}(0),g^{\mathrm{\prime }}(0),h^{\mathrm{\prime }}(0),q^{\mathrm{\prime }}(0)>0$, and $\underset{s\to \mathrm{\infty }}{\lim }\frac{f(Mg(s))}{s}=0$ 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.