Classification of Radial Solutions for Semilinear Elliptic Systems with Nonlinear Gradient Terms

TL;DR

This paper classifies positive radial solutions for a semilinear elliptic system with nonlinear gradient terms, establishing integral conditions for existence and analyzing solution behavior using dynamical systems.

Contribution

It introduces an integral criterion analogous to Keller-Osserman for the existence of solutions in systems with gradient nonlinearities and describes their asymptotic behavior.

Findings

Solutions exist if and only if a specific integral diverges.

Derived optimal conditions for solutions in the whole space and bounded domains.

Analyzed solution behavior for power nonlinearities using dynamical systems.

Abstract

We are concerned with the classification of positive radial solutions for the system $\Delta u=v^p$, $\Delta v=f(|\nabla u|)$, where $p>0$ and $f\in C^1[0,\infty)$ is a nondecreasing function such that $f(t)>0$ for all $t>0$. We show that in the case where the system is posed in the whole space $\mathbb{R}^N$ such solutions exist if and only if $\displaystyle\int_{1}^\infty \Big(\displaystyle \int_0^s F(t)dt \Big)^{-p/(2p+1)} ds =\infty$. This is the counterpart of the Keller-Osserman condition for the case of single semilinear equation. Similar optimal conditions are derived in case where the system is posed in a ball of $\mathbb{R}^N$. If $f(t)=t^q$, $q>1$, using dynamical system techniques we are able to describe the behaviour of solutions at infinity (in case where the system is posed in the whole $\mathbb{R}^N$) or around the boundary (in case of a ball).