Asymptotically homogeneous solutions of the supercritical Lane-Emden system

We consider the Lane-Emden system $$\begin{aligned} -\Delta u = \vert v\vert ^{p-1}v, \quad -\Delta v= \vert u\vert ^{q-1}u\qquad \text{ in } \; \mathbb {R}^d, \; { d\ge 2}. \end{aligned}$$ When $$p\ge q\ge 1,$$ it is known that there exists a positive radial stable solution $$(u,v)\in C^2(\mathbb R^d)\times C^2(\mathbb R^d)$$ if and only if $$d\ge 11$$ and (p, q) lies on or above the so-called Joseph-Lundgren curve introduced in Chen, Dupaigne and Ghergu (Discrete Continent Dynamic System 34, 2469-2479, 2014). 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 1.2 below). Most of our results are optimal improvements of previous works in the literature.