Descripción: In this work we consider the nonlocal stationary nonlinear problem (J * u)(x) - u(x) = -λu(x) + a(x)up(x) in a domain Ω, with the Dirichlet boundary condition u(x) = 0 in ℝN Ω and p > 1. The kernel J involved in the convolution (J * u)(x) = ∫ℝN J(x - y)u(y) dy is a smooth, compactly supported nonnegative function with unit integral, while the weight a(x) is assumed to be nonnegative and is allowed to vanish in a smooth subdomain Ω0 of Ω. Both when a(x) is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter λ.

Descripción: In this paper we consider the elliptic system Δ u = a (x) up vq, Δ v = b (x) ur vs in Ω, a smooth bounded domain, with boundary conditions frac(∂ u, ∂ ν) = λ u, frac(∂ v, ∂ ν) = μ v on ∂Ω. Here λ and μ are regarded as parameters and p, s > 1, q, r > 0 verify (p - 1) (s - 1) > q r. 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 over(Ω, -). 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 = ∞. © 2009 Elsevier Inc. All rights reserved.