Lane–Emden systems with negative exponents

We study the elliptic system{−Δu=u−pv−qin Ω,−Δv=u−rv−sin Ω,u=v=0on ∂Ω, in a bounded domain Ω⊂RNΩ⊂RN (N⩾1N⩾1) with a smooth boundary, p,s⩾0p,s⩾0 and q,r>0q,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 C1C1-regularity of solutions up to the boundary is also obtained.