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

Abstract

We are concerned with the classification of positive radial solutions for the system $\hat{I}”u=v^p$, $\hat{I}”v=f(|\nabla u|)$, 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 ${\int }_1^{\infty }({\int }_0^{s}F(t)dt{)}^{-p/(2p+1)}ds=\infty$. 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).

29 pages, 3 figures