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We consider the Lane-Emden system-$\Delta$u = |v| p-1 v,-$\Delta$v = |u| q-1u in R d. When p $\ge$ q $\ge$ 1, it is known that there exists a positiveradial 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]. Inthis 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 criticalSobolev hyperbola), while for d $\ge$ 11, the Joseph-Lundgren curve is indeedthe dividing line for the existence of such solutions, if one assumes inaddition that they are asymptotically homogeneous (see Definition 1 below).Most of our results are optimal improvements of previous works in thelitterature.

Asymptotically homogeneous solutions of the supercritical Lane-Emden system