An elliptic system with bifurcation parameters on the boundary conditions

In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)⩾0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω0, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ1⩽∞, μ>0, where λ1 is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ1<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ1=∞.