Lane–Emden systems with negative exponents

We study the elliptic system ⎧⎨⎩ − u = u−pv−q in Ω, − v = u−r v−s in Ω, u = v =0 on ∂Ω, in a bounded domain Ω ⊂ RN (N 1) with a smooth boundary, p, s 0 and q, r > 0. We investigate the existence, non-existence, and uniqueness of C2(Ω)∩C(Ω) solutions in terms of p, q, r and s. A necessary and sufficient condition for the C1-regularity of solutions up to the boundary is also obtained. © 2010 Elsevier Inc. All rights reserved