Blow-up radial solutions for elliptic systems with monotonic non-linearities

We are concerned with the existence and boundary behaviour of positive radial solutions for the system \begin{equation*} \left{ \begin{aligned} Δu&=g(|x|,v(x)) &&\quad\mbox{in}\ Ω, \ Δv&=f(|x|,|\nabla u(x)|) &&\quad\mbox{in}\ Ω, \end{aligned} \right. \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is either a ball centered at the origin or the whole space ${\mathbb{R}}^N$, and $f,g\in C^1([0,\mathrm{\infty })\times [0,\mathrm{\infty }))$, are non-negative, and increasing. Firstly, we study the existence of positive radial solutions in the case when the system is posed in a ball corresponding to their behaviour at the boundary. Next, we discuss the existence of positive radial solutions in case when $g(\mathrm{\mid }x\mathrm{\mid },v(x))=\mathrm{\mid }x{\mathrm{\mid }}^a{v}^p$ and $f(\mathrm{\mid }x\mathrm{\mid },\mathrm{\mid }\mathrm{\nabla }u(x)\mathrm{\mid })=\mathrm{\mid }x{\mathrm{\mid }}^{b}h(\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })$. Finally, we take $h(t)=t^s$, $s>1$, $\mathrm{\Omega }={\mathbb{R}}^N$ and by the use of dynamical system techniques we are able to describe the behaviour at infinity of such positive radial solutions.