We study the elliptic system −Δ u = δ( x ) −a v p in Ω, −Δ v = δ( x ) −b u q in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝ N , N ≥ 1, is a smooth and bounded domain, δ( x ) = dist( x , ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q .
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Marius Ghergu (2011) studied this question.