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

We are concerned with non-constant positive radial solutions of the system $$\{\begin{array}{rlrl}{\displaystyle S_k(D^{2}u)}& {\displaystyle =\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^m{v}^p}& {\displaystyle }& {\displaystyle in\mathrm{\Omega },}\\ {\displaystyle S_k(D^{2}v)}& {\displaystyle =\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q{v}^s}& {\displaystyle }& {\displaystyle in\mathrm{\Omega },}\end{array}$$ where $S_k(D^{2}u)$ is the $k$-Hessian operator of $u\in C^2(\Omega)$($1\leq k\leq N$)and$\Omega\subset$\mathbb{R}$^N$ $(N\ge 2)$ is either a ball or the whole space. The exponents satisfy $q>0$, $m,s\ge 0$, $p\ge s\ge 0$ and $(k-m)(k-s)\mathrm{\ne }pq$. In the case where $\mathrm{\Omega }$ is a ball, we classify all the positive radial solutions according to their behavior at the boundary. Further, we consider the case $\mathrm{\Omega }={\mathbb{R}}^N$ and find that the above system admits non-constant positive radial solutions if and only if $0\le m<k$ 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.