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 { − [ M 1 ( ∫ Ω | ∇ u | p ) ] p − 1 Δ p u = f ( u , v ) + ρ 1 ( x ) in  Ω , − [ M 2 ( ∫ Ω | ∇ v | p ) ] p − 1 Δ p v = g ( u , v ) + ρ 2 ( x ) in  Ω , ∂ u ∂ η = ∂ v ∂ η = 0 on  ∂ 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 W 1 , 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 W 1 , p ( Ω ) formed by the functions with null mean in Ω . In this way, and motivated by physical motivations related to wave equation we consider the conditions ( F 1 ) – ( F 2 ) .