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} Δu&=g(|x|,v(x)) &&\quad\mbox{in}\ Ω, \\ Δv&=f(|x|,|\nabla u(x)|) &&\quad\mbox{in}\ Ω, \end{aligned} \right. \end{equation*} where $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ 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 $h(t)=t^s$, $s>1$, $\hat{I}\text{c◯}={\mathbb{R}}^N$ and by the use of dynamical system techniques we are able to describe the behaviour at infinity of such positive radial solutions.

21 pages, substantial text overlap with arXiv:2206.01868