Blow-up radial solutions for elliptic systems with monotonic non-linearities

Abstract

We are concerned with the existence and boundary behaviour of positive radial solutions for the system \begin{equation*} \left\{ \begin{aligned} \Delta u&=g(|x|,v(x)) &&\quad\mbox{in}\ \Omega, \\ \Delta v&=f(|x|,|\nabla u(x)|) &&\quad\mbox{in}\ \Omega, \end{aligned} \right. \end{equation*} where $\Omega \subset \mathbb{R}^N$ is either a ball centered at the origin or the whole space $\mathbb{R}^N$, and $f,g\in C^{1}([0,\infty)\times [0,\infty))$, are non-negative, and increasing. Firstly, we study the existence of positive radial solutions in the case when the system is posed in a ball corresponding to their behaviour at the boundary. Next, we discuss the existence of positive radial solutions in case when $g(|x|,v(x)) = |x|^{a} v^p$ and $f(|x|, |\nabla u (x)|) = |x|^{b} h(|\nabla u|)$. Finally, we take $h(t) = t^s$, $s> 1$, $\Omega = \mathbb{R}^N$ and by the use of dynamical system techniques we are able to describe the behaviour at infinity of such positive radial solutions.