Existence of positive radial solution for the elliptic system on an exterior domain

This article discusses the existence of positive radial solution for the elliptic system $$ \left \{ \textstyle\begin{array}{l} -\triangle u=K(|x|)f(|x|,\ u,\ v,\ |\nabla u|),~~x\in \Omega , \\ -\triangle v=K(|x|)g(|x|,\ u,\ v,\ |\nabla v|),~~x\in \Omega , \\ \alpha _{1}u+\beta _{1}\frac{\partial u}{\partial n}|_{\partial \Omega}=0,~\alpha _{2}v+\beta _{2}\frac{\partial v}{\partial n}|_{ \partial \Omega}=0, \\ \lim _{|x|\to \infty} u(x)=0,~\lim _{|x|\to \infty} v(x)=0, \end{array}\displaystyle \right . $$ where $\Omega =\{x\in \mathbb{R}^{N}:~|x|>r_{0}>0\}$ , $N\ge 3$ , $K:[r_{0},~ \infty )\to \mathbb{R}_{+}$ , f and $g:[r_{0},~\infty )\times \mathbb{R}_{+}^{3}\to \mathbb{R}_{+}$ are continuous, $\mathbb{R}_{+}=[0,~ \infty )$ . Under the correlation conditions of the nonlinear terms $f(r, \ u,\ v,\ \xi )$ and $g(r,\ u,\ v,\ \eta )$ may be super-linear or sub-linear growth on u, v, ξ, and η, existence results of positive radial solutions are obtained. For the super-linear growth case, Nagumo condition (F3) is presented to restrict the growth of $f(r,~u,~v,~\xi )$ and $g(r,~u,~v,~ \eta )$ on ξ and η. The super-linear growth or sub-linear growth of the nonlinear terms f and g are described by the correlation inequality conditions instead of the usual the independent limits conditions about f and g. The discussion is based on the fixed point index theory on cones.