On a nonlocal elliptic system of p-Kirchhoff-type under Neumann boundary condition

Ω |∇u| )]p−1 ∆pu = f (u, v)+ ρ1(x) in Ω, − [ M2 (∫ Ω |∇v| )]p−1 ∆pv = g(u, v)+ ρ2(x) in Ω, ∂u ∂η = ∂v ∂η = 0 on ∂Ω, (1.1) where Ω ⊂ R,N ≥ 1, is a bounded smooth domain, 1 < p < N, η is the unit exterior vector on ∂Ω , ∆p is the p-Laplacian operator ∆pu = div(|∇u|p−2∇u) ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (F.J.S.A. Corrêa), [email protected] (R.G. Nascimento). 0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.03.013 F.J.S.A. Corrêa, R.G. Nascimento / Mathematical and Computer Modelling 49 (2009) 598–604 599 and the involved functions in the problem satisfy: (M)  M1,M2 : R → R are continuous functions and there is a positive constant m0 > 0 such that M1(t),M2(t) ≥ m0 > 0, for all t ≥ 0; (F1) { There is a C1 − function F : R2 → R such that Fu(u, v) = f (u, v) and Fv(u, v) = g(u, v) for all (u, v) ∈ R2; (F2) There is k > 0 such that F(u+ k, v+ k) = F(u, v), for all (u, v) ∈ R2; (ρ) ρ1,ρ2 ∈ L (Ω), 1 p + 1 q = 1, 1 < q < p and ∫