Global and blow-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form $$\{\begin{array}{rlrl}{\displaystyle {\mathrm{\Delta }}_{p}u}& {\displaystyle =v^m\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{\alpha }}& {\displaystyle }& {\displaystyle in\mathrm{\Omega },}\\ {\displaystyle {\mathrm{\Delta }}_{p}v}& {\displaystyle =v^{\beta }\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q}& {\displaystyle }& {\displaystyle in\mathrm{\Omega },}\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ $(N\ge 2)$ is either a ball or the whole space, $1<p<\mathrm{\infty }$, $m,q>0$, $\alpha \ge 0$, $0\le \beta \le m$ and $(p-1-\alpha )(p-1-\beta )-qm\mathrm{\ne }0$. We first classify all the positive radial solutions in case $\mathrm{\Omega }$ is a ball, according to their behavior at the boundary. Then we obtain that the system has non-constant global solutions if and only if $0\le \alpha <p-1$ and $mq< (p-1-\alpha)(p-1-\beta)$. Finally, we describe the precise behavior at infinity for such positive global radial solutions by using properties of three component cooperative and irreducible dynamical systems.