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We study positive radial solutions of quasilinear elliptic systems with agradient term in the form $$ \left\{ \begin{aligned} \Delta_{p} u&=v^{m}|\nablau|^{\alpha}&&\quad\mbox{ in }\Omega,\\ \Delta_{p} v&=v^{\beta}|\nabla u|^{q}&&\quad\mbox{ in }\Omega, \end{aligned} \right. $$ where $\Omega\subset\R^N$$(N\geq 2)$ is either a ball or the whole space, $1<p<\infty$, $m, q>0$,$\alpha\geq 0$, $0\leq \beta\leq m$ and $(p-1-\alpha)(p-1-\beta)-qm\neq 0$. Wefirst classify all the positive radial solutions in case $\Omega$ is a ball,according to their behavior at the boundary. Then we obtain that the system hasnon-constant global solutions if and only if $0\leq \alpha<p-1$ and $mq<(p-1-\alpha)(p-1-\beta)$. Finally, we describe the precise behavior at infinityfor such positive global radial solutions by using properties of threecomponent cooperative and irreducible dynamical systems.

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