In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δu = vp, Δv = uq in Ω, where p, q > 0, under the boundary condition u|∂Ω = +∞, v|∂Ω = +∞. It is shown that no positive solutions exist when pq ≤ 1, while we provide a detailed account of the set of (infinitely many) positive solutions if pq > 1. The behaviour near the boundary of all solutions is also elucidated, and symmetric solutions (u, v) are completely characterized in terms of their minima (u(0), v(0)). Non-symmetric solutions are also deeply studied in the one-dimensional problem.