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

In this paper we investigate questions of existence of solution for the system {−[M1(∫Ω|∇u|p)]p−1Δpu=f(u,v)+ρ1(x)in Ω,−[M2(∫Ω|∇v|p)]p−1Δpv=g(u,v)+ρ2(x)in Ω,∂u∂η=∂v∂η=0on ∂Ω. Motivated by a problem in [D.G. Costa, Tópicos em análise funcional não-linear e aplicações às equações diferenciais, VIII Escola Latino-Americana de Matemática, Rio de Janeiro, Brazil, 1986. [3]], who studies a single local equation, we study the above problem by using variational methods. Since we will work in the space W1,p(Ω)W1,p(Ω), the functional associated to the above problem will not be coercive. So, we have to consider the Poincaré–Wirtinger’s inequality in the subspace of W1,p(Ω)W1,p(Ω) formed by the functions with null mean in ΩΩ. In this way, and motivated by physical motivations related to wave equation we consider the conditions (F1)–(F2)(F1)–(F2).