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 $$ \left\{ \begin{aligned} Δ_{p} u&=v^{m}|\nabla u|^α&&\quad\mbox{ in }Ω,\\ Δ_{p} v&=v^β|\nabla u|^{q} &&\quad\mbox{ in }Ω, \end{aligned} \right. $$ where $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ $(N\ge 2)$ is either a ball or the whole space, $1<p<\infty$, $m,q>0$, $\hat{I}\pm \ge 0$, $0\le {\hat{I}}^2\le m$ and $(p-1-\hat{I}\pm )(p-1-{\hat{I}}^2)-qm\ne 0$. We first classify all the positive radial solutions in case $\hat{I}\text{c◯}$ 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 $0\le \hat{I}\pm <p-1$ and $mq<(p-1-\hat{I}\pm )(p-1-{\hat{I}}^2)$. 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.

21 pages, 2 figures