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

Abstract

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form $$ \left\{ \begin{aligned} \Delta_{p} u&=v^{m}|\nabla u|^{\alpha}&&\quad\mbox{ in }\Omega,\\ \Delta_{p} v&=v^{\beta}|\nabla u|^{q} &&\quad\mbox{ in }\Omega, \end{aligned} \right. $$ 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 $(p-1-\alpha)(p-1-\beta)-qm\neq 0$. We first 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 has non-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 infinity for such positive global radial solutions by using properties of three component cooperative and irreducible dynamical systems.