Classification of radial solutions for elliptic systems driven by the $k$-Hessian operator

Abstract

We are concerned with non-constant positive radial solutions of the system $$ \left\{ \begin{aligned} S_k(D^2 u)&=|\nabla u|^{m} v^{p}&&\quad\mbox{ in }\Omega,\\ S_k(D^2 v)&=|\nabla u|^{q} v^{s} &&\quad\mbox{ in }\Omega, \end{aligned} \right. $$ where $S_k(D^2u)$ is the $k$-Hessian operator of $u\in C^2(\Omega)$ ($1\leq k\leq N$) and $\Omega\subset\mathbb{R}^N$ $(N\geq 2)$ is either a ball or the whole space. The exponents satisfy $q>0$, $m,s\geq 0$, $p\geq s\geq 0$ and $(k-m)(k-s)\neq pq$. In the case where $\Omega$ is a ball, we classify all the positive radial solutions according to their behavior at the boundary. Further, we consider the case $\Omega=\mathbb{R}^N$ and find that the above system admits non-constant positive radial solutions if and only if $0\leq m<k$ and $pq < (k-m)(k-s)$. Using arguments from three component cooperative and irreducible dynamical systems we deduce the behavior at infinity of such solutions.