The solvability of an elliptic system under a singular boundary condition

Source abstract

In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δ u = v p , Δ v = u q 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.