  Let $\Omega\subset\mathbb R^{n}\ (n\geq2)$ be either an open ball $B_R$
centred at the origin or the whole space. We study the existence of positive,
radial solutions of quasilinear elliptic systems of the form \begin{equation*}
\left\{ \begin{aligned} \Delta_{p} u&=f_1(|x|)g_1(v)|\nabla u|^{\alpha}
&&\quad\mbox{ in } \Omega, \\ \Delta_{p} v&=f_2(|x|)g_2(v)h(|\nabla u|)
&&\quad\mbox{ in } \Omega, \end{aligned} \right. \end{equation*} where
$\alpha\geq 0$, $\Delta_{p}$ is the $p$-Laplace operator, $p>1$, and for
$i,j=1,2$ we assume $f_i,g_j,h$ are continuous, non-negative and non-decreasing
functions. For functions $g_j$ which grow polynomially, we prove sharp
conditions for the existence of positive radial solutions which blow up at
$\partial B_{R}$, and for the existence of global solutions.
