Existence and boundary behaviour of radial solutions for weighted elliptic systems with gradient terms

TL;DR

This paper investigates the existence and boundary behavior of positive radial solutions for a weighted elliptic system with gradient terms, using dynamical systems techniques to analyze solutions in both bounded domains and the whole space.

Contribution

It provides new results on the existence and boundary behavior of solutions for weighted elliptic systems with gradient terms, including the case of entire solutions with polynomial growth.

Findings

Existence of positive radial solutions in bounded domains.

Boundary behavior characterized for solutions near the boundary.

Asymptotic behavior of solutions at infinity in the whole space.

Abstract

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