  We consider the Lane-Emden system-$\Delta$u = |v| p-1 v,-$\Delta$v = |u| q-1
u in R d. When p $\ge$ q $\ge$ 1, it is known that there exists a positive
radial stable solution (u, v) $\in$ C 2 (R d) if and only if d $\ge$ 11 and (p,
q) lies on or above the so-called Joseph-Lundgren curve introduced in [5]. In
this paper, we prove that for d $\le$ 10, there is no positive stable solution
(or merely stable outside a compact set and (p, q) does not lie on the critical
Sobolev hyperbola), while for d $\ge$ 11, the Joseph-Lundgren curve is indeed
the dividing line for the existence of such solutions, if one assumes in
addition that they are asymptotically homogeneous (see Definition 1 below).
Most of our results are optimal improvements of previous works in the
litterature.
