An elliptic system with bifurcation parameters on the boundary conditions

• Autores: García-Melián, J.; Rossi, J.D.; Sabina de Lis, J.C.
• Año de publicación: 2009
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: J. Differ. Equ. 2009;247(3):779-810
• Materia: Asymptotic profiles; Bifurcation; Elliptic semilinear systems of competitive type; Sub- and supersolutions
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

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.
Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.