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

TL;DR

This paper classifies positive radial solutions of a quasilinear elliptic system with gradient terms, determining conditions for existence of global solutions and describing their behavior at the boundary or infinity.

Contribution

It provides a complete classification of solutions for the system, including existence criteria and asymptotic behavior, extending understanding of such elliptic systems in radial symmetry.

Findings

All positive radial solutions in a ball are classified by boundary behavior.

Global solutions exist if and only if specific parameter conditions are met.

The asymptotic behavior of solutions at infinity is characterized using dynamical systems analysis.

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.